FRAGMENTS OF ALMOST RING THEORY

LORENZO RAMERO

Contents

1. Introduction 1 2. Homological theory 2 2.1. Almost categories 2 2.2. Almost homological algebra 6 2.3. Almost homotopical algebra 9 2.4. Almost cotangent complex 15 3. Almost ring theory 16 3.1. Flat, unramified and étale morphisms 16 3.2. Lifting theorems 19 References 23

1. Introduction

Let V be a ring and 𝔪 an ideal of V such that 𝔪^{2} = 𝔪. The V -modules killed by 𝔪 are the object of a (full) Serre subcategory Σ of the category V − Mod of all V -modules, and the quotient category V − Mod/Σ is an abelian category V − al.Mod which we call the category of almost V -modules. It is easy to check that the usual tensor product of V -modules descends to a bifunctor ⊗ on almost V -modules, so that V − al.Mod is a monoidal abelian category in a natural way. Then an almost ring is just an almost V -module A endowed with a “multiplication” morphism A ⊗ A → A satisfying certain natural axioms. Together with the obvious morphisms, these gadgets form a category V −al.Alg and there is a natural localisation functor V − Alg → V − al.Alg which associates to any V -algebra the same object viewed in the localised category.

While the notion of almost V -module had already arisen (in the sixties) in Gabriel’s memoir “Des categories abeliennes” [2], the usefulness of almost rings did not become apparent until Faltings’ paper [1] on “p-adic Hodge theory”. To be accurate, the definition of almost étale extension found in [1] is still given in terms of usual rings and modules, and the idea of passing to the quotient category is not really developed, rather it is scattered around in a series of clues that an honest reader may choose to pursue if so inclined.

About two years ago I decided that, if no one else was interested in writing up this story, I could as well take a stab at it myself. The result of my efforts appeared in the preprint [7]. The main aim of [7] was to provide convenient foundations for the study of deformations of almost rings and their morphisms. In view of applications to p-adic Hodge theory, it is especially interesting to establish suitable almost versions of the standard results on infinitesimal lifting of étale rings and morphisms. The method I chose to tackle this question, was to extend Illusie’s theory of the cotangent complex to the almost case. This method worked reasonably well, and yielded more general results than could be achieved with Hochschild cohomology (Faltings’ original method). As an added bonus, proofs using the cotangent complex also require far fewer calculations.

Last year, on invitation of Francesco Baldassarri, I presented these results in a talk in Padova. Due to time constraints, I preferred to forgo the detailed introduction of almost terminology, and rather to focus on deformations. That is, I dealt throughout the lecture with usual rings and modules but I gave proofs of the lifting theorems which could be adapted to the almost case with relative ease (to obtain such proofs, one has just to copy the arguments given in [7], omitting everywhere the word “almost”). After the talk, it was convened that I would write up my lecture as a contribution to the Rendiconti. This would have had to be a mainly expository article, and much of the work would have gone into fleshing out the references to homotopical algebra, so that the paper could have also been useful as a first introduction to the theory of the cotangent complex.

However, a string of circumstances kept on delaying this project in the intervening months. In the meanwhile, Ofer Gabber sent me some important remarks about my preprint; these remarks led to extensive discussions and eventually to the decision of collaborating on a radical overhaul of the preprint. Our new approach is based on the systematic exploitation of the left adjoint V − al.Alg → V − Alg to the localization functor. This approach is in many ways superior to the previous one (which was based on the right adjoint); in particular, rather than extending Illusie’s theory to the almost case, one can now reduce the study of deformations of almost rings to the case of usual rings.

The net outcome of these developments is that, on one hand the theory of deformations of almost rings is now almost as polished as the theory for usual rings, and on the other hand, much of the industrious work that buttressed the first preprint has been obsoleted.

Since I am still rather fond of the elementary arguments that I devised for the first version, I prefer to think instead that this material has been liberated from the paper and is now free to be included in a revised note for Rendiconti. The following article is a compromise born out of these considerations : it is at the same time more and less than what was originally planned. It is more, because it now consists mostly of original results. It is less, because it is no longer designed as a gentle first introduction to the cotangent complex and to general homotopical algebra; besides, whenever the proof of a result has not significantly changed from the first to the second preprint, I avoid duplication and refer instead to the appropriate statement in the “official” version [3].

2. Homological theory

2.1. Almost categories. Unless otherwise stated, every ring is commutative with unit. If C is a category, and X, Y two objects of C, we will usually denote by Hom_{C}(X, Y) the set of morphisms in C from X to Y and by 1_{X} the identity morphism of X. Moreover we denote by C^{o} the opposite category of C and by s.C the category of simplicial objects over C, that is, functors Δ^{o} → C, where Δ is the category whose objects are the ordered sets [n] = {0, ..., n} for each integer n ≥ 0 and where a morphism ϕ : [p] → [q] is a non-decreasing map. A morphism f : X → Y in s.C is a sequence of morphisms f_{[n]} : X[n] → Y [n], n ≥ 0 such that the obvious diagrams commute. We can imbed C in s.C by sending each object X to the “constant” object s.X such that s.X[n] = X for all n ≥ 0 and s.X[ϕ] = 1_{X} for all morphisms ϕ in Δ. We let Δ(n) be the simplicial set represented by [n], i.e such that Δ(n)[i] = Hom_{Δ}([i], [n]).

If C is an abelian category, D(C) will denote the derived category of C. As usual we have also the subcategories D^{+}(C), D^{−}(C) of complexes of objects of C which are exact for sufficiently large negative (resp. positive) degree. If A is a ring, the category of A-modules (resp. Aalgebras) will be denoted by A − Mod (resp. A − Alg). Most of the times we will write Hom_{A}(M, N) instead of Hom_{A−Mod}(M, N).

We denote by Set the category of sets and by Ab the category of abelian groups. The symbol ℕ denotes the set of non-negative integers; in particular 0 ∈ ℕ.

Our basic setup consists of a fixed base ring V containing an ideal 𝔪 such that 𝔪^{2} = 𝔪. However, pretty soon we will need to introduce further hypotheses on the pair (V, 𝔪); such additional restrictions are conveniently expressed by the following axioms (S1) and (S2). Before discussing them, let us stress that many interesting assertions can be proved without assuming either of these axioms, and some more require only the validity of (S1), which is the weakest of the two. For these reasons, neither of them is part of our basic setup, and they will be explicitly invoked only as the need arises.

(S1) The ideal 𝔪 is generated by a multiplicative system S ⊂ 𝔪 with the following property. For all x ∈ S there exist y, z ∈ S such that x = y^{2} · z.

Suppose that axiom (S1) holds. The divisibility relation induces a partial order on S by setting y ≺ x if and only if there exists z ∈ S ∪ {1} such that x = y · z.

(S2) The ideal 𝔪 is generated by a multiplicative system S ⊂ 𝔪 consisting of non-zero divisors and the resulting partially ordered set (S, ≺) is cofiltered.

Let us remark that (S2) implies (S1). First of all, without loss of generality we can assume that u · x ∈ S whenever u ∈ V is a unit and x ∈ S. Next, pick any x ∈ S; if x is smaller (under ≺) than every element of S, then 𝔪 = x · V. Then 𝔪 = 𝔪^{2} implies that x^{2} divides x, and the claim follows easily. Otherwise, there exists s ∈ S which is not greater than x; then (S2) implies that there exists t, t′ ∈ S such that x = t · t′; again (S2) gives an element y ∈ S which divides both t and t′, so y^{2} divides x and (S1) follows.

Moreover, the second axiom implies that 𝔪 is a flat V -module : indeed, under (S2), 𝔪 is a filtered colimit of the free V -modules s · V, where s ranges over all the elements of S.

Example 2.1.1. i) The main example is given by a non-discrete valuation ring V with valuation ν : V −{0} → Γ of rank one (where Γ is the totally ordered abelian group of values of ν). Then we can take for S the set of elements x ∈ V − {0} such that ν(x) > ν(1) and (S2) is satisfied.

ii) Suppose that S contains invertible elements. This is the “classical limit”. In this case almost ring theory reduces to usual ring theory. Thus, all the discussion that follows specialises to, and sometimes gives alternative proofs for, statements about rings and their modules.

Let M be a given V -module. We say that M is almost zero if 𝔪 · M = 0. A morphism ϕ of V -modules is an almost isomorphism if both Ker(ϕ) and Coker(ϕ) are almost zero V -modules.

The full subcategory Σ of V −Mod consisting of all V -modules which are almost isomorphic to 0 is clearly a Serre subcategory and hence we can form the quotient category V − al.Mod = V − Mod/Σ (after taking some set-theoretic precautions : the interested reader will find in [8] (§10.3) a discussion of these issues). Then V − al.Mod is an abelian category which will be called the category of almost V -modules.

There is a natural localization functor

V − Mod → V − al.Mod M ↦ M^{a}

which takes a V -module M to the same module, seen as an object of V − al.Mod.

Since the almost isomorphisms form a multiplicative system (see e.g. [8] Ex. 10.3.2), it is possible to describe the morphisms in V − al.Mod via a calculus of fractions, as follows. Let V − al.Iso be the category which has the same objects as V − Mod, but such that Hom_{V −al.Iso}(M, N) consists of all almost isomorphisms M → N. If M is any object of V − al.Iso we write (V − al.Iso/M) for the category of objects of V − al.Iso over M (i.e. morphisms ϕ : X → M). If ϕ_{i} : X_{i} → M (i = 1, 2) are two objects of (V − al.Iso/M) then Hom_{(V −al.Iso/M)}(ϕ_{1}, ϕ_{2}) consists of all morphisms ψ : X_{1} → X_{2} in V − al.Iso such that ϕ_{1} = ϕ_{2} ∘ ψ. For any two V -modules M, N we define a functor F_{N} : (V − al.Iso/M)^{o} → V − Mod by associating to an object ϕ : P → M the V -module Hom_{V} (P, N) and to a morphism α : P → Q the map

Hom_{V} (Q, N) → Hom_{V} (P, N) : β ↦ β ∘ α.

Then we have

(2.1.2) Hom_{V −al.Mod}(M^{a}, N^{a}) = colim_{(V −al.Iso/M)^{o}} F_{N}.

In particular Hom_{V −al.Mod}(M, N) has a natural structure of V -module, for any two almost V -modules M, N, i.e. Hom_{V −al.Mod}(−, −) is a bifunctor which takes values in the category V − Mod.

The usual tensor product induces a bifunctor − ⊗_{V} − on almost V -modules, which, in the jargon of [5] makes of V − al.Mod a (closed symmetric) monoidal category. Then an almost V -algebra is just a monoid in the monoidal category V − al.Mod. This means (a bit sloppily) the following. For two almost V -modules M and N let η_{M|N} : M ⊗_{V} N → N ⊗_{V} M be the automorphism which “switches the two factors” (m ⊗ n ↦ n ⊗ m) and ν_{A/V} : V ⊗_{V} A → A the “scalar multiplication” (v ⊗ a ↦ va). Then an almost V -algebra is an almost V -module A endowed with a morphism of almost V -modules μ_{A/V} : A ⊗_{V} A → A (the “multiplication” of A) and a “unit morphism” \underline{1}_{A/V} : V → A satisfying the conditions

μ_{A/V} ∘ (1_{A} ⊗_{V} μ_{A/V}) = μ_{A/V} ∘ (μ_{A/V} ⊗_{V} 1_{A}) (associativity)
μ_{A/V} = μ_{A/V} ∘ η_{A|A}  (commutativity)
μ_{A/V} ∘ (\underline{1}_{A/V} ⊗_{V} 1_{A}) = ν_{A/V}  (unit property)

With the morphisms defined in the obvious way, the almost V -algebras form a category V − al.Alg. Clearly the localization functor restricts to a functor V − Alg → V − al.Alg. Occasionally we will have to deal with non-commutative or non-unital almost algebras; for these more general monoids, the second, resp. the third of the above axioms fails.

Furthermore, a left almost A-module is an almost V -module M endowed with a morphism σ_{M} : A ⊗_{V} M → M such that

\begin{matrix}σ_{M} ∘ (1_{A} ⊗_{V} σ_{M}) = & σ_{M} ∘ (μ_{A/V} ⊗_{V} 1_{M}) \\ σ_{M} ∘ (\underline{1}_{A/V} ⊗_{V} 1_{M}) = & ν_{M/V} . \end{matrix} (2.1.3)

Similarly we define right almost A-modules. The A-linearity of a morphism ϕ : M → N of A-modules is expressed by the condition

σ_{N} ∘ (1_{A} ⊗_{V} ϕ) = ϕ ∘ σ_{M}. (2.1.4)

We denote by A − al.Mod the category of left almost A-modules and A-linear morphisms defined as one expects. Clearly A − al.Mod is an abelian category. For any V -algebra B we have a localization functor B − Mod → B^{a} − al.Mod.

Next, if A is an almost V -algebra, we can define the category A − al.Alg of almost Aalgebras. It consists of all the morphisms A → B of almost V -algebras.

Being, as they are, objects in a quotient category, almost A-modules do not possess elements in the same way as usual modules do. However, not everything is lost, as we show in the following definition.

Definition 2.1.5. Let M be an almost A-module. An almost A-element of M is just an A-linear morphism A → M. We denote by M_{∗} the set of all almost A-elements of the almost A-module M. If B is an almost A-algebra, we can multiply almost elements as follows. First we remark that the morphism μ_{A/A} : A ⊗_{A} A → A is an isomorphism. Then, given almost elements b_{1} : A → B and b_{2} : A → B set b_{1} · b_{2} = μ_{B/A} ∘ (b_{1} ⊗_{A} b_{2}) ∘ μ_{A/A}^{−1} : A → B. In particular, A_{∗} is a V -algebra and B_{∗} is endowed with a natural A_{∗}-algebra structure, whose identity is the structure morphism \underline{1}_{B/A} : A → B.

Moreover, if m : A → M_{1} is an almost A-element of M_{1} and ϕ : M_{1} → M_{2} is an Alinear morphism, we denote by ϕ_{∗}(m) : A → M_{2} the almost A-element of M_{2} defined as ϕ_{∗}(m) = ϕ ∘ m. In this way we obtain an A_{∗}-linear morphism ϕ_{∗} : M_{1}_{∗} → M_{2}_{∗}, i.e. the assignment M ↦ M_{∗} extends to a functor from almost A-modules (resp. almost A-algebras) to A_{∗}-modules (resp. A_{∗}-algebras).

For any two almost A-modules M, N, the set Hom_{A−al.Mod}(M, N) has a natural structure of A_{∗}-module and we obtain an internal Hom functor by letting

alHom_{A}(M, N) = Hom_{A−al.Mod}(M, N)^{a}.

This is the functor of almost homomorphisms from M to N.

For any almost A-module M we have also a functor of tensor product M ⊗_{A} − on almost A-modules which, in view of the following proposition 2.1.7 can be shown to be a left adjoint to the functor alHom_{A}(M, −). It can be defined as M ⊗_{A} N = (M_{∗} ⊗_{A_{∗}} N_{∗})^{a} but an appropriate almost version of the usual construction would also work.

With this tensor product, A − al.Mod is a monoidal category as well, and A − al.Alg could also be described as the category of monoids in the category of almost A-modules. Under this equivalence, a morphism ϕ : A → B of V -algebras becomes the unit morphism \underline{1}_{B/A} : A → B of the corresponding monoid. We will sometimes drop the subscript and write simply \underline{1}.

Suppose that B_{1} and B_{2} are two almost A-algebras and M is an almost B_{1}-module. Then we can induce a structure of almost B_{2} ⊗_{A} B_{1}-module on B_{2} ⊗_{A} M, by declaring that the scalar multiplication σ_{B_{2}⊗_{A}M} : (B_{2} ⊗_{A} B_{1}) ⊗_{A} (B_{2} ⊗_{A} M) → B_{2} ⊗_{A} M acts by the rule: (c_{1} ⊗ b) ⊗ (c_{2} ⊗ m) ↦ (c_{1} · c_{2}) ⊗ (b · m) (for all c_{1}, c_{2} ∈ B_{2∗}, b ∈ B_{1∗}, m ∈ M_{∗}). In categorical notation, this translates as follows. Let η_{B_{1}|B_{2}} : B_{1} ⊗_{A} B_{2} → B_{2} ⊗_{A} B_{1} be the A-linear isomorphism which “switches the two factors”; we set

σ_{B_{2}⊗_{A}M} = (μ_{B_{2}/A} ⊗_{A} σ_{M}) ∘ (1_{B_{2}} ⊗_{A} η_{B_{1}|B_{2}} ⊗_{A} 1_{M}). (2.1.6)

Proposition 2.1.7. i) There is a natural isomorphism A ≃ A_{∗}^{a} of almost V -algebras.

ii) The functor M ↦ M_{∗} from A−al.Mod to A_{∗}−Mod (resp. from A−al.Alg to A_{∗}−Alg) is right adjoint to the localization functor A_{∗} − Mod → A_{∗}^{a} − al.Mod = A − al.Mod (resp. A_{∗} − Alg → A − al.Alg).

iii) The counit of the adjunction M_{∗}^{a} → M is a natural equivalence from the composition of the two functors to the identity functor 1_{A−al.Mod} (resp. 1_{A−al.Alg}).

Proof. (i) : quite generally, for any almost A-module M we have a standard isomorphism of A_{∗}-modules

(2.1.8) Hom_{A−al.Mod}(A, M) ≃ Hom_{V −al.Mod}(V, M).

Then the claim follows easily from the explicit description of the set of morphisms in V − al.Mod given above.

(ii) : For any given almost A_{∗}^{a}-module N and any A_{∗}-module M we need to establish a natural bijection

(2.1.9) Hom_{A_{∗}^{a}−al.Mod}(M^{a}, N) ≃ Hom_{A_{∗}}(M, N_{∗}).

For any m ∈ M define the morphism of A_{∗}-modules σ_{m} : A_{∗} → M by a ↦ a · m. Then the bijection sends a morphism ϕ : M^{a} → N of almost A_{∗}^{a}-modules to the morphism of A_{∗}- modules ϕ̃ : M → N_{∗} defined as m ↦ ϕ ∘ σ_{m}^{a}. Moreover, ϕ̃ is a morphism of A_{∗}^{a}-algebras whenever ϕ is a morphism of almost A-algebras.

An explicit inverse for the bijection can be obtained as follows. Pick any A_{∗}-linear morphism ϕ̃ : M → N_{∗}. We construct a morphism of V -modules ϕ : 𝔪 · M → N/(𝔪 − torsion) which represents the required morphism of almost A_{∗}^{a}-modules M^{a} → N. If m ∈ M, by definition the morphism ϕ̃(m) ∘ \underline{1}_{A/V} : V → N is represented by a pair consisting of an almost isomorphism ψ : X → V and a V -linear morphism ϕ′ : X → N. For given ε ∈ 𝔪 choose u ∈ X such that ε = ψ(u) and define ϕ(ε · m) to be the class modulo 𝔪-torsion of ϕ′(u). We leave it to the reader to verify that ϕ thus defined extends to a unique A_{∗}^{a}-linear map and that the assignment ϕ̃ ↦ ϕ is an inverse for the map (2.1.9).

For (iii) we need to show that the natural morphism ϕ : M_{∗}^{a} → M corresponding via (2.1.9) to the identity of M_{∗}, is an isomorphism of almost A-modules. Inspecting the proof of (ii), we see that this morphism is represented by a V -linear morphism 𝔪·M_{∗} → M/(𝔪−torsion) which can be explicitly computed. We indicate an inverse for ϕ and leave the details to the reader. For m ∈ M, let ω_{m} : A → A ⊗_{V} M be the A-linear morphism a ↦ a ⊗ m. Then the morphism M → M_{∗}^{a} defined as m ↦ σ_{M} ∘ ω_{m} provides the required inverse.

Remark 2.1.10. (i) Proposition 2.1.7 follows also directly from [2] (chap.III §3 Cor.1).

(ii) It is also easy to check that, for any V -module M, the natural map (unit of the adjunction) M → M_{∗}^{a} is an almost isomorphism. Moreover, under (S2) we have a natural isomorphism

M_{∗}^{a} ≃ Hom_{V} (𝔪, M) ≃ lim_{ε∈S} M

where, for any two elements ε ≺ δ of the cofiltered set S, the corresponding morphism ϕ_{ε,δ} : M → M is defined by m ↦ (δ/ε) · m.

Corollary 2.1.11. The categories A − al.Mod and A − al.Alg are both complete and cocomplete.

Proof. We recall that the categories A_{∗}−Mod and A_{∗}−Alg are both complete and cocomplete. Now let I be any small indexing category and M : I → A − al.Mod be any functor. Denote by M_{∗} : I → A_{∗} − Mod the composed functor i ↦ M(i)_{∗}. We claim that

colim_{I} M = (colim_{I} M_{∗})^{a}.

The proof is an easy application of proposition 2.1.7(iii). A similar argument also works for limits and for the category A − al.Alg.

Next recall that the forgetful functor A_{∗} − Alg → Set (resp. A_{∗} − Mod → Set) has a left adjoint A_{∗}[−] : Set → A_{∗} − Alg (resp. A^{(−)} : Set → A_{∗} − Mod) which assigns to a set S the free A_{∗}-algebra A_{∗}[S] (resp. the free A_{∗}-module A_{∗}^{(S)}) generated by S. If S is any set, it is natural to write A[S] (resp. A^{(S)}) for the almost A-algebra (A_{∗}[S])^{a} (resp. for the almost A-module (A_{∗}^{(S)})^{a}. This yields a left adjoint, called the free almost algebra functor Set → A−al.Alg (resp. the free almost module functor Set → A−al.Mod) to the “forgetful” functor A − al.Alg → Set (resp. A − al.Mod → Set) B ↦ B_{∗}.

Remark 2.1.12. The functor of almost elements commutes with arbitrary limits, because all right adjoints do. It does not in general commute with arbitrary colimits, not even with arbitrary infinite direct sums.

2.2. Almost homological algebra. In this section we fix an almost V -algebra A and we consider various constructions in the category of almost A-modules.

Remark 2.2.1. i) Let M_{1}, M_{2} be almost A-modules. By proposition 2.1.7 it is clear that a morphism ϕ : M_{1} → M_{2} of almost A-modules is uniquely determined by the induced morphism M_{1}_{∗} → M_{2}_{∗}.

ii) It is a bit tricky to deal with preimages of almost elements under morphisms: for instance, if ϕ : M_{1} → M_{2} is an epimorphism (by which we mean that Coker(ϕ) ≃ 0) and m_{2} ∈ M_{2}_{∗}, then it is not true in general that we can find an almost A-element m_{1} ∈ M_{1}_{∗} such that ϕ_{∗}(m_{1}) = m_{2}. What remains true is that for arbitrary ε ∈ 𝔪 we can find m_{1} such that ϕ_{∗}(m_{1}) = ε · m_{2}.

Suppose that (S1) holds. Let T = {t_{n} | n ∈ ℕ} be an infinite sequence of elements of S. We say that T is a Cauchy sequence if, for any ε ∈ S there exists s = s(ε) ∈ S such that s ≺ t_{n} ≺ s · ε for all but finitely many n ∈ ℕ. We say that T converges to 1 if, for any s ∈ S, we have t_{n} ≺ s for all but finitely many n ∈ ℕ. We illustrate the use of this language in the proof of the following lemma.

Lemma 2.2.2. Assume (S2) and let {M_{n} ; ϕ_{n} : M_{n} → M_{n+1} | n ∈ ℕ} (resp. {N_{n} ; ψ_{n} : N_{n+1} → N_{n} | n ∈ ℕ}) be a direct (resp. inverse) system of almost A-modules and morphisms and {ε_{n} | n ∈ ℕ} a sequence of S which converges to 1.

i) If ε_{n} · M_{n} = 0 for all n ∈ ℕ then colim_{n∈ℕ}M_{n} ≃ 0.

ii) If ε_{n} · N_{n} = 0 for all n ∈ ℕ then lim_{n∈ℕ}N_{n} ≃ 0 ≃ lim_{n∈ℕ}^{1}N_{n}.

iii) If ε_{n} · Coker(ψ_{n}) = 0 for all n ∈ ℕ and moreover {∏_{j=0}^{n} ε_{j} | n ∈ ℕ} is a Cauchy sequence, then lim_{n∈ℕ}^{1}N_{n} ≃ 0.

Proof. (i) and (ii) are left as exercises to the reader. We prove (iii). Let ψ be the morphism

ψ : ∏_{n=0}^{∞} N_{n} → ∏_{n=0}^{∞} N_{n}

which assigns to any sequence (b_{n} | n ∈ ℕ) of almost A-elements b_{n} ∈ N_{n}_{∗} the sequence

ψ_{∗}(b_{n} | n ∈ ℕ) = (ψ_{n}_{∗}(b_{n+1}) − b_{n} | n ∈ ℕ).

It is a standard result (see e.g. [8] (§3.5)) that lim_{n∈ℕ}^{1}N_{n} ≃ Coker(ψ). In other words, we

have lim_{n∈ℕ}^{1}N_{n} ≃ (lim_{n∈ℕ}^{1}N_{n}_{∗})^{a}. Let δ ∈ S be any element; it suffices therefore to show that

δ · lim_{n∈ℕ}^{1}N_{n}_{∗} = 0.

After replacing ε_{n} by ε_{n}^{2} we can assume that ε_{n} · Coker(ψ_{n}_{∗}) = 0 for all n ∈ ℕ. Moreover, for any fixed m ∈ ℕ we have a natural isomorphism lim_{n∈ℕ}^{1}N_{n}_{∗} ≃ lim_{n∈ℕ}^{1}N_{n+m}_{∗}. Hence, up to omitting the first m modules and renumbering the others, we can assume that

∏_{j=0}^{n} ε_{j} ≺ δ for all n ∈ ℕ.

Set δ_{n} = δ/ ∏_{j=0}^{n−1} ε_{j} for all n ∈ ℕ. Also, for any element (c_{n} | n ∈ ℕ) of ∏_{n=0}^{∞} N_{n}_{∗} we denote by (c_{n}^{∗} | n ∈ ℕ) the new element defined by c_{n}^{∗} = δ_{n} · c_{n}.

Now, let (c_{n} | n ∈ ℕ) be any almost A-element of ∏_{n=0}^{∞} N_{n}_{∗}. We will construct inductively a sequence (a_{n} | n ∈ ℕ) with the property that c_{n}^{∗} + a_{n}^{∗} = ψ_{n}_{∗}(a_{n+1}^{∗}) for all n ∈ ℕ, in other words ψ_{∗}(a_{n}^{∗} | n ∈ ℕ) = (c_{n}^{∗} | n ∈ ℕ).

We let a_{0} = 0. Suppose that we have already found a_{1}, ..., a_{n} such that c_{j}^{∗} + a_{j}^{∗} = ψ_{j}_{∗}(a_{j+1}^{∗}) for all j < n. Since ε_{n} · Coker(ψ_{n}_{∗}) = 0, we can find a_{n+1} ∈ N_{n+1}_{∗} such that ε_{n} · (c_{n} + a_{n}) = ψ_{n}_{∗}(a_{n+1}). We multiply both sides of this equation by δ_{n+1} to obtain

δ_{n} · (c_{n} + a_{n}) = δ_{n+1} · ψ_{n}_{∗}(a_{n+1})

which is what we need. Finally, for any sequence (b_{n} | n ∈ ℕ) as above, let (c_{n} | n ∈ ℕ) be the sequence defined by c_{n} = ∏_{j=0}^{n−1} ε_{j} · b_{n} for all n ∈ ℕ. It is clear that (c_{n}^{∗} | n ∈ ℕ) = δ · (b_{n} | n ∈ ℕ), which means that δ · Coker(ψ) = 0, as required.

Definition 2.2.3. Let M be an almost A-module.

i) We say that M is flat if the functor N ↦ M ⊗_{A} N, from the category of almost A-modules to itself is exact. M is almost projective if the functor N ↦ alHom_{A}(M, N) is exact.

ii) We say that M is almost finitely generated if, for arbitrarily small ε ∈ S there exist a positive integer n = n(ε) and an A-linear morphism ϕ_{ε} : A^{n} → M such that ε · Coker(ϕ_{ε}) = 0.

iii) We say that M is almost finitely presented if, for arbitrarily small ε ∈ S there exist positive integers n = n(ε), m = m(ε) and a three term complex A^{m} →^{ψ_{ε}} A^{n} →^{ϕ_{ε}} M such that ε · Coker(ϕ_{ε}) = ε · Ker(ϕ_{ε})/Im(ψ_{ε}) = 0.

The abelian category A − al.Mod satisfies axiom (AB5) (see e.g. [8] (§A.4)) and it has a generator, namely the object A itself. It then follows by a general result that A − al.Mod has enough injectives. It is also clear that A − al.Mod has enough almost projective (resp. flat) objects. Given an almost A-module M, we can derive the functors M ⊗_{A}− (resp. alHom_{A}(M, −), resp. alHom_{A}(−, M)) by taking flat (resp. injective, resp. almost projective) resolutions (one remarks that bounded above exact complexes of flat (resp. almost projective) almost modules are acyclic for the functor M ⊗_{A} − (resp. alHom_{A}(−, M)), and then uses the construction detailed in [8] th. 10.5.9). We denote by Tor_{i}^{A}(M, −) (resp. alExt_{A}^{i}(M, −), resp. alExt_{A}^{i}(−, M)) the corresponding derived functors. If A = B^{a} for some V -algebra B we obtain easily natural isomorphisms

Tor_{i}^{B}(M, N)^{a} ≃ Tor_{i}^{A}(M^{a}, N^{a})

for all B-modules M, N. A similar result holds for Ext_{B}^{i}(M, N).

Remark 2.2.4. i) Clearly, an almost A-module M is flat (resp. almost projective) if and only if Tor_{i}^{A}(M, N) = 0 (resp. alExt_{A}^{i}(M, N) = 0) for all almost A-modules N and all i > 0.

ii) Let M, N be two flat (resp. almost projective) almost A-modules. Then M ⊗_{A} N is a flat (resp. almost projective) A-module and for any almost B-algebra A, the almost B-module B ⊗_{A} M is flat (resp. almost projective).

Lemma 2.2.5. (see [3]) Let M be an almost finitely generated almost A-module. Consider the following properties:

i) M is almost projective.

ii) For arbitrary ε ∈ 𝔪 there exist n(ε) ∈ ℕ and A-linear morphisms

\xymatrix{M \ar[r]^{u_{ε}} & A^{n(ε)} \ar[r]^{v_{ε}} & M} (2.2.6)

such that v_{ε} ∘ u_{ε} = ε · 1_{M}.

iii) M is flat.

Then (i) ⇔ (ii) ⇒ (iii).

There is a converse to lemma 2.2.5 in case M is almost finitely presented. To prove it, we need some preparation.

Let R be a ring and M any R-module. Recall (see [8] (3.2.3)) that the Pontrjagin dual of M is the R-module M^{†} = Hom_{Ab}(M, ℚ/ℤ). An element r of R acts on M^{†} via (r·f)(m) = f(r·m) (for f ∈ M^{†}, m ∈ M).

Proposition 2.2.7. (cp. [8] §3.2)

i) The Pontrjagin dual is a contravariant exact functor on the category R − Mod.

ii) A sequence M → N → P of morphisms of R-modules is exact if and only if the dual sequence P^{†} → N^{†} → M^{†} is exact.

iii) For any two R-modules M, N there is a natural R-linear morphism σ : M^{†} ⊗_{R} N → Hom_{R}(N, M)^{†}, defined by σ(f ⊗ n) : h ↦ f(h(n)), which is an isomorphism if N is finitely presented.

Corollary 2.2.8. Let M be a finitely presented R-module and r ∈ R an element such that r · Tor_{i}^{R}(M, N) = 0 for any R-module N and any integer i > 0. Then we have also r · Ext_{R}^{i}(M, N) = 0 for any R-module N and any integer i > 0.

Proof. Suppose we are given a surjection ϕ : B → C of R-modules. It suffices to show that r kills the cokernel of the induced morphism ϕ_{∗} : Hom_{R}(M, B) → Hom_{R}(M, C). By proposition 2.2.7(i) the map C^{†} → B^{†} is injective. We consider the commutative diagram

\xymatrix{C^{†} ⊗_{R} M \ar[r] \ar[d] & B^{†} ⊗_{R} M \ar[d] \\ Hom_{R}(M, C)^{†} \ar[r] & Hom_{R}(M, B)^{†}}

whose vertical arrows are isomorphisms according to 2.2.7(iii). By hypothesis, r kills the kernel of the upper horizontal arrows, hence also the kernel of the lower one. By proposition 2.2.7(i) this kernel is the Pontrjagin dual of D = Coker(ϕ_{∗}) and by proposition 2.2.7(ii) it follows easily that r must kill D itself, as required.

Proposition 2.2.9. If (S1) holds then every almost finitely presented flat almost A-module is almost projective.

Proof. Let M be such an almost A-module. For ε ∈ S pick morphisms A^{m} →^{ψ} A^{n} →^{ϕ} M such that ε · Coker(ϕ) = ε · (Ker(ϕ)/Im(ψ)) = 0. Let C = Coker(ψ_{∗} : A_{∗}^{m} → A_{∗}^{n}). The kernel and cokernel of the natural morphism C → M_{∗} are killed by ε^{2}. Moreover, by hypothesis, ε kills Tor_{i}^{A_{∗}}(M_{∗}, N) for all A_{∗}-modules N and all i > 0. It follows easily that ε^{5} kills Tor_{i}^{A_{∗}}(C, N) for all A_{∗}-modules N and all i > 0. Then it follows from corollary 2.2.8 that ε^{5} also kills Ext_{A_{∗}}^{i}(C, N) for all A_{∗}-modules N and all i > 0. In turns, this implies that ε^{9} kills Ext_{A_{∗}}^{i}(M_{∗}, N) for all A_{∗}-modules N and all i > 0. As ε can be taken arbitrarily small, the claim follows.

For the proof of the following two lemmata, we refer the reader to [3].

Lemma 2.2.10. Assume (S2) and let {M_{n} ; ϕ_{n} : M_{n} → M_{n+1} | n ∈ ℕ} be a direct system of almost A-modules and suppose there exist sequences T = {ε_{n} | n ∈ ℕ} and {δ_{n} | n ∈ ℕ} of elements of S such that

i) T converges to 1 and {∏_{j=0}^{n} δ_{j} | n ∈ ℕ} is a Cauchy sequence;

ii) for all n ∈ ℕ there exist integers N(n) and morphisms of almost A-modules ψ_{n} : A^{N(n)} → M_{n} such that ε_{n} · Coker(ψ_{n}) = 0;

iii) δ_{n} · Coker(ϕ_{n}) = 0 for all n ∈ ℕ.

Then colim_{n∈ℕ}M_{n} is an almost finitely generated almost A-module.

Lemma 2.2.11. Assume (S2) and let {M_{n} ; ϕ_{n} : M_{n} → M_{n+1} | n ∈ ℕ} be a direct system of almost A-modules and suppose there exist sequences T = {ε_{n} | n ∈ ℕ} and {δ_{n} | n ∈ ℕ} of elements of S such that

i) T converges to 1 and P = {∏_{j=0}^{n} δ_{j} | n ∈ ℕ} is a Cauchy sequence;

ii) ε_{n} · alExt_{A}^{i}(M_{n}, N) = δ_{n} · alExt_{A}^{i}(Coker(ϕ_{n}), N) = 0 for all almost A-modules N, all i > 0 and all n ∈ ℕ;

iii) δ_{n} · Ker(ϕ_{n}) = 0 for all n ∈ ℕ.

Then colim_{n∈ℕ}M_{n} is an almost projective almost A-module.

2.3. Almost homotopical algebra. Our first task is to extend all of section 2.1 to simplicial almost modules and algebras. This requires no particular effort, so we only sketch how to proceed. A simplicial almost V -algebra is just an object in the category s.(V − al.Alg). Then, for a given simplicial algebra A, we define the category A − al.Mod of almost A-modules : it consists of all simplicial almost V -modules M such that M[n] is an almost A[n]-module and such that the face and degeneracy morphisms d_{i} : M[n] → M[n−1] and s_{i} : M[n] → M[n+1] (i = 0, 1, ..., n) are A[n]-linear. For instance, if A is an almost V -algebra, we can form the constant simplicial almost V -algebra s.A and then the category s.A − al.Mod is the same thing as the category s.(A − al.Mod) of simplicial almost A-modules. Sometimes we may have to consider, for a given simplicial almost algebra A, the category s.(A − al.Mod) of simplicial almost A-modules. This is the same as the category of all bisimplicial complexes of almost V -modules, with additional A[n]-linearity conditions which the reader can easily figure out. The category A − al.Mod is abelian and it is even a monoidal category with the tensor product formed dimension-wise: (M ⊗_{A} N)[n] = M[n] ⊗_{A[n]} M[n]. The internal hom functor s.alHom_{A}(−, −) : A − al.Mod^{o} × A − al.Mod → A − al.Mod can be defined as

s.alHom_{A}(X, Y)[n] = Hom_{A−al.Mod}(X ⊗_{V} V^{Δ(n)}, Y)^{a}

with face morphisms induced naturally from those of Δ(n). Here V^{Δ(n)} denotes the simplicial almost V -module obtained by applying dimension-wise the free almost module functor to the simplicial set Δ(n).

Again, an almost A-algebra B is a monoid in the category of almost A-modules, which is the same as a simplicial almost V -algebra with a morphism of simplicial almost V -algebras A → B. We let A − al.Alg be the category of these objects.

Next, by applying dimension-wise the functor of almost elements we may form the simplicial V -algebra A_{∗}. The functor of almost elements then extends to a functor A − al.Mod → A_{∗} − Mod (and similarly for almost algebras). Then the whole of proposition 2.1.7 and its corollary carries over to this more general situation.

Similarly we have forgetful functors A_{∗} − Mod → s.Set (resp. A_{∗} − Alg → s.Set) and their adjoints associate to a simplicial set S the simplicial free module (resp.algebra) generated dimension-wise by S[n]. These can be composed with the previous functors to give constructions of free simplicial almost A-modules A^{(S)} and A-algebras A[S].

In the following we let A be some fixed simplicial almost V -algebra. Our next task is to extend parts of [4] to the almost case, leading up to the construction of the almost cotangent complex in the coming section. Most of the work consists in poring over Illusie’s manuscript and accepting that the arguments given there carry through verbatim to the almost setting. This is usually the case, with few exceptions mostly due to the fact that the functor of almost elements is only left exact (rather than exact). We will point out these potential difficulties as they arise.

Definition 2.3.1. Let B be any almost A-algebra and M any almost B-module. An almost A-derivation of B with values in M is an A-linear morphism ∂ : B → M such that

∂_{[n]∗}(b_{1} · b_{2}) = b_{1} · ∂_{[n]∗}(b_{2}) + b_{2} · ∂_{[n]∗}(b_{1})

for all objects [n] of Δ and all b_{1}, b_{2} ∈ B_{∗}[n]. The set of all M-valued almost A-derivations of B forms a V -module Der_{A}(B, M) and the corresponding almost V -module alDer_{A}(B, M) has a natural structure of almost B[0]-module (via the degeneracy morphisms).

Definition 2.3.2. Let again B be an almost A-algebra. An ideal of B is a monomorphism I → B of almost B-modules. We reserve the notation I_{B/A} for the ideal Ker(μ_{B/A} : B ⊗_{A} B → B). The almost module of relative differentials of ϕ is defined as the (left) almost B-module Ω_{B/A} = I_{B/A}/I_{B/A}^{2}. It is endowed with a natural almost A-derivation

δ : B → Ω_{B/A}

which is defined by b ↦ \underline{1} ⊗ b − b ⊗ \underline{1} for all b ∈ B_{∗}. The assignment (A → B) ↦ Ω_{B/A} defines a functor

Ω : s.al.Alg.Morph → s.al.Alg.Mod

from the category of morphisms A → B of simplicial almost V -algebras to the category denoted s.al.Alg.Mod consisting of all pairs (B, M) where B is a simplicial almost V -algebra and M is an almost B-module. The morphisms in s.al.Alg.Morph are all the commutative squares; the morphisms (B, M) → (B′, M′) in s.al.Alg.Mod are all pairs (ϕ, f) where ϕ : B → B′ is a morphism of simplicial almost V -algebras and f : B′ ⊗_{B} M → M′ is a morphism of almost B′-modules.

The almost module of relative differentials enjoys the familiar universal properties which one expects. In particular Ω_{B/A} represents the functor Der_{A}(B, −), i.e. for any (left) almost B-module M the morphism

(2.3.3)  Hom_{B−al.Mod}(Ω_{B/A}, M) → Der_{A}(B, M)  f ↦ f ∘ δ

is an isomorphism. As an exercise, the reader can supply the proof for this claim and for the following standard proposition.

Proposition 2.3.4. i) Let B and C two almost A-algebras. Then there is a natural isomorphism Ω_{C⊗_{A}B/C} ≃ C ⊗_{A} Ω_{B/A}.

ii) Let B be an almost A-algebra and C an almost B-algebra. Then there is a natural exact sequence of almost C-modules

C ⊗_{B} Ω_{B/A} → Ω_{C/A} → Ω_{C/B} → 0.

iii) Let I be an ideal of the almost A-algebra B and let C = B/I be the quotient almost A-algebra. Then there is a natural exact sequence

I/I^{2} → C ⊗_{B} Ω_{B/A} → Ω_{C/A} → 0. (2.3.5)

iv) The functor Ω : s.al.Alg.Morph → s.al.Alg.Mod commutes with all colimits.

Definition 2.3.6. An A-extension of an almost A-algebra B by an almost B-module I is a short exact sequence of almost A-modules

\xymatrix{X :  0 \ar[r] & I \ar[r] & C \ar[r]^{p} & B \ar[r] & 0}

such that C is an almost A-algebra, p is a morphism of almost A-algebras and I is a square zero ideal in C. The A-extensions form a category Exal_{A}. The morphisms are commutative diagrams with exact rows

\xymatrix{X : \ar[d] & 0 \ar[r] & I \ar[r] \ar[d]^{f} & E \ar[r]^{p} \ar[d]^{g} & B \ar[r] \ar[d]^{h} & 0 \\ X′ : & 0 \ar[r] & I′ \ar[r] & E′ \ar[r]^{p′} & B′ \ar[r] & 0}

such that g and h are morphisms of almost A-algebras. We denote by Exal_{A}(B, I) the subcategory of Exal_{A} consisting of all A-extensions of B by I, where the morphisms are all short exact sequences as above such that f = 1_{I} and h = 1_{B}.

For a morphism ϕ : C → B of almost A-algebras, and an A-extension X in Exal_{A}(B, I), we can pullback X via ϕ to obtain an A-extension X ∗ ϕ in Exal_{A}(C, I) with a morphism X ∗ ϕ → X of A-extensions. Similarly, given a B-linear morphism ψ : I → J, we can push out X and obtain an object ψ ∗ X in Exal_{A}(B, J) with a morphism X → ψ ∗ X of Exal_{A}. These operations can be used to induce a structure of abelian group on the set Exal_{A}(B, I) of isomorphism classes of objects of Exal_{A}(B, I) as follows. For any two objects X, Y of Exal_{A}(B, I) we can form X⊕Y which is an object of Exal_{A}(B⊕B, I⊕I). Let ϕ : B → B⊕B be the diagonal morphism and ψ : I ⊕ I → I the addition morphism of I. Then we set X + Y = ψ ∗ (X ⊕ Y) ∗ ϕ. One can check that X + Y ≃ Y + X for any X, Y and that the trivial split A-extension B ⊕ I is a neutral element for +. Moreover every isomorphism class has an inverse −X. The functors X ↦ X ∗ ϕ and X ↦ ψ ∗ X commute with the operation thus defined, and induce group homomorphisms

∗ϕ : Exal_{A}(B, I) → Exal_{A}(C, I) and ψ∗ : Exal_{A}(B, I) → Exal_{A}(B, J).

Definition 2.3.7. (cp. [4] (III.1.1.7)) We say that an almost A-algebra C verifies condition (L) if, for all A-extensions X = (0 → I → B → C → 0), the sequence of C-modules

0 → I → C ⊗_{B} Ω_{B/A} → Ω_{C/A} → 0 (2.3.8)

obtained by extending (2.3.5)by zero on the left, is exact.

Suppose now that C verifies (L). Then we denote by diff(X) the exact sequence (2.3.8) associated to the A-extension X. This defines a functor from Exal_{A}(C, I) to the category of extensions (in the category of almost C-modules) of Ω_{C/A} by I. Hence we derive a morphism of abelian groups

diff : Exal_{A}(C, I) → Ext_{C−al.Mod}^{1}(Ω_{C/A}, I)

where Ext^{1} denotes here the Yoneda Ext functor on the abelian category of almost C-modules.

Conversely, let C be any almost A-algebra and Y = (0 → I → J → Ω_{C/A} → 0) an exact sequence of almost C-modules. We deduce an A-extension of (C ⊗_{A} C)/I_{C/A}^{2} ≃ C ⊕ Ω_{C/A}

Y′ : 0 → I → C ⊕ J → C ⊕ Ω_{C/A} → 0

where J is also a square zero ideal in C ⊕ J and I is an almost C ⊕ Ω_{C/A}-module annihilated by the ideal Ω_{C/A}. Let j_{2} : C → (C ⊗_{A} C)/I_{C/A}^{2} be the morphism c ↦ 1 ⊗ c (c ∈ C_{∗}). Then we obtain an A-extension of C by I

alg(Y) = Y′ ∗ j_{2}.

This defines a group homomorphism

(2.3.9) alg : Ext_{C−al.Mod}^{1}(Ω_{C/A}, I) → Exal_{A}(C, I).

The basic observation is the following:

Proposition 2.3.10. (cp. [4] (III.1.1.9)) Suppose that the almost A-algebra C verifies condition (L). Then the homomorphisms diff and alg are inverse to each other.

Our aim is to extend proposition 2.3.10 to general almost A-algebras. The strategy will be the following: find a suitable (functorial) simplicial resolution P → C by a system of A-algebras P [n] satisfying condition (L). Then form the simplicial C-module 𝕃_{C/A} = Ω_{P/A} ⊗_{P} C. This will replace the C-module Ω_{C/A} in (2.3.9). Moreover the Yoneda Ext will be replaced by a hyperext functor, computed in a suitable derived category. This program will occupy the rest of this section and the following one. To start out, we verify that there are enough A-algebras to play with, which satisfy (L).

Lemma 2.3.11. Assume that (S2) holds and suppose that C = A[S] for some simplicial set S. Then C satisfies (L).

Proof. This is one of the points which require a little extra care. Let X = (0 → I → B →^{f} C → 0) be an A-extension. We need to show that the sequence (2.3.8) is exact. By virtue of proposition 2.3.4(iv) we can assume that S is a simplicial finite set (i.e. all S[n] are finite sets). By applying termwise the functor of almost elements to X we obtain a sequence X_{∗}. However the morphism f_{∗} will not be in general surjective, but rather only almost surjective. Now pick ε ∈ S and consider the morphism ϕ_{ε} : A_{∗}[S] → A[S]_{∗} which sends a generator s ∈ S[n] ⊂ A_{∗}[S[n]] to the element ε · s ∈ A[S[n]]_{∗}. Let X_{ε∗} = X_{∗} ∗ ϕ_{ε}. Then X_{ε∗} is an A_{∗}-extension of A_{∗}[S] by I. According to [4] (III.1.1.7.1) the almost A_{∗}-algebra A_{∗}[S] satisfies the analogous condition (L) for A_{∗}-algebras. Hence we can form the exact sequence of A_{∗}[S]- modules diff(X_{ε∗}). Let D_{ε} = diff(X_{ε∗})^{a} be the exact sequence of A[S]-modules obtained by applying termwise the functor M ↦ M^{a}. Because of (S2) the sequences D_{ε} form a direct system for varying ε ∈ S and moreover the colimit of the system is naturally isomorphic to the sequence (2.3.8). As a filtered colimit of exact sequences of almost modules is again exact, the claim follows.

Next, we construct a few auxiliary derived categories, for the future complex 𝕃_{B/A} to live in. A bit more generally, let C be any abelian category. Recall (see [4] (I.1.3)) the construction of the normalized complex which associates to every object X of s.C a chain complex N(X)^{•} defined by

N(X)_{n} = ∩_{i>0}Ker(d_{i} : X[n] → X[n − 1]). (2.3.12)

(We have used the standard convention N(X)_{n} = N(X)^{−n}). It is an object of the category C_{•}(C) of chain complexes M^{•} of objects of C such that M^{i} = 0 for all i > 0. The theorem of Dold-Kan (see [8] th.8.4.1) states that X ↦ N(X) induces an equivalence

N : s.C → C_{•}(C)

and in fact an explicit essential inverse to N can be produced. Now we say that a morphism X → Y in s.C is a quasi-isomorphism if the induced morphism of N(X) → N(Y) is a quasiisomorphism of chain complexes.

Definition 2.3.13. Let A be a simplicial almost V -algebra. We say that a morphism ϕ : M → N of almost A-modules (or almost A-algebras) is a quasi-isomorphism if the morphism ϕ of underlying simplicial almost V -modules is a quasi-isomorphism. We define the category D_{•}(A) (resp. D_{•}(A − al.Alg)) as the localization of the category A − al.Mod (resp. A − al.Alg) with respect to the class of quasi-isomorphisms. (We assume that our universe is large enough to accommodate this kind of constructions).

If A is an almost V -algebra, then the functor N induces an equivalence from D_{•}(s.A) to the derived category D_{•}(A − al.Mod) which is the localization of C_{•}(A − al.Mod) with respect to the class of quasi-isomorphisms. Many tools that are available for derived categories have adequate simplicial counterparts. For instance we have a suspension functor Σ : A−al.Mod → A − al.Mod (for any simplicial V -algebra A). We recall the definition from [4] (I.3.2.1). Let \xymatrix{d_{0}, d_{1} : Δ(0) \ar[r] \ar[r] & Δ(1)} be the two natural simplicial maps and set σ = V^{Δ(1)}/(ImV^{(d_{0})} + ImV^{(d_{1})}). This is a simplicial almost V -module whose normalised chain complex is

N(σ) = (0 → V → 0)

where the component V is placed in homological degree 1. Then set ΣM = σ ⊗_{V} M for any almost A-module M. It follows from the Eilenberg-Zilber theorem (see [8] th.8.5.1) that the simplicial almost V -module underlying N(ΣM) decomposes as a direct sum K(M)⊕N(M)[1] where K(M) is a contractible complex depending functorially on M (and where F ↦ F [1] denotes the usual shift operator on the category C_{•}(V − al.Mod)). Similarly we have a cone functor on almost A-modules, defined as E ↦ C(E) = γ ⊗_{V} E where γ is the simplicial V - module V^{Δ(1)}/ImV^{(d_{0})}. The morphism d_{1} : Δ(0) → Δ(1) induces a morphism i_{E} : E → C(E) and there is a short exact sequence

\xymatrix{0 \ar[r] & E \ar[r]^{i_{E}} & C(E) \ar[r]^{p_{E}} & ΣE \ar[r] & 0.}

With this notation we can now introduce the hyperext functor which will replace Yoneda Ext. It is defined as

𝔼xt_{A}^{p}(E, F) = colim_{n≥−p} Hom_{D_{•}(A)}(Σ^{n}E, Σ^{n+p}F)

for any two almost A-modules E, F and for any integer p. It turns out that the computation of 𝔼xt^{1} can be reduced to some extent, to the computation of certain related Yoneda Ext^{1} groups. We are going to describe how this is accomplished.

First, for a morphism u : E → F of almost A-modules we define the cone C(u) via the push-out diagram

\xymatrix{E \ar[r]^{i_{E}} \ar[d]^{u} & C(E) \ar[d] \\ F \ar[r] & C(u).}

We derive a sequence of morphisms of almost A-modules

\xymatrix{E \ar[r]^{u} & F \ar[r] & C(u) \ar[r]^{p_{u}} & ΣE} (2.3.14)

where the morphism p_{u} : C(u) → ΣE is induced, via the universal property of the push-out, by the morphism p_{E} and the zero morphism F → ΣE.

Definition 2.3.15. We call a triangle of D_{•}(A) any sequence of morphisms in D_{•}(A)

L → M → N → ΣL.

A morphism from a triangle (L → M → N → ΣL) to a triangle (L′ → M′ → N′ → ΣL′) is a sequence of morphisms f : L → L′, g : M → M′ and h : N → N′ such that the diagram

\xymatrix{L \ar[r] \ar[d]^{f} & M \ar[r] \ar[d]^{g} & N \ar[r] \ar[d]^{h} & ΣL \ar[d]^{Σf} \\ L′ \ar[r] & M′ \ar[r] & N′ \ar[r] & ΣL′}

commutes. We say that a triangle is distinguished if it is isomorphic to a triangle of the kind (2.3.14).

Now let X = (0 → E →^{u} F → G → 0) be a short exact sequence of almost A-modules. One shows (see [4] (I.3.2.3)) that X induces a natural quasi-isomorphism s_{X} : C(u) → G and moreover the triangle

\xymatrix{E \ar[r]^{u} & F \ar[r] & G \ar[r]^{p_{u}∘s_{X}^{−1}} & ΣE}

is distinguished. We set

χ(X) = (p_{u} ∘ s_{X}^{−1} : G → ΣE) ∈ Hom_{D_{•}(A)}(G, ΣE).

It is easy to check that χ(X) depends only on the class of X in the Yoneda Ext^{1} group and moreover, for any morphisms f : E → E′, g : G′ → G we have

χ(f ∗ X) = (Σf) ∘ χ(X) χ(X ∗ g) = χ(X) ∘ g

which means that χ induces a group homomorphism

χ : Ext_{A−al.Mod}^{1}(G, E) → 𝔼xt_{A}^{1}(G, E)

functorial in both variables E, G. Finally we have the following

Proposition 2.3.16. (cp. [4] (I.3.2.3.8)) Let E, F be two almost A-modules.

i) for any u ∈ 𝔼xt_{A}^{1}(E, F) there is a quasi-isomorphism t : F → F′ (resp. s : E′ → E) and an extension X (resp. Y) of E by F′ (resp. of E′ by F) such that (Σt)^{−1} ∘ χ(X) = u (resp. χ(Y) ∘ s^{−1} = u).

ii) Let t′ : F → F′ (resp; t′′ : F → F′′) be a quasi-isomorphism and X′ (resp. X′′) an extension of E by F′ (resp. F′′). We have (Σt′)^{−1} ∘ χ(X′) = (Σt′′)^{−1} ∘ χ(X′′) if and only if there exist quasi-isomorphisms r′ : F′ → G and r′′ : F′′ → G such that r′ ∘ t′ is homotopic to r′′ ∘ t′′ and r′ ∗ X′ = r′′ ∗ X′′ in Ext_{A−al.Mod}^{1}(E, G).

iii) Let s′ : E′ → E (resp. s′′ : E′′ → E) be a quasi-isomorphism and Y′ (resp. Y′′) an extension of E′ (resp. E′′) by F. We have χ(Y′) ∘ (s′)^{−1} = χ(Y′′) ∘ (s′′)^{−1} if and only if there exist quasi-isomorphisms r′ : D → E′ and r′′ : D → E′′ such that s′ ∘ r′ is homotopic to s′′ ∘ r′′ and Y′ ∗ r′ = Y′′ ∗ r′′ in Ext_{A−al.Mod}^{1}(D, F).

2.4. Almost cotangent complex. Let A → B be a morphism of almost V -algebras. In this section we complete the program announced in section 2.3 : first of all we construct a natural simplicial resolution of B by almost A-algebras which satisfy condition (L) of definition 2.3.7. This is just the simplicial almost A-algebra P_{A}(B) = P_{A_{∗}}(B_{∗})^{a} where P_{A_{∗}}(B_{∗}) is the simplicial A_{∗}-algebra associated to the morphism A_{∗} → B_{∗} as in [4] (II.1.2.1). It comes with a natural augmentation

P = P_{A}(B) →^{p} B

which induces a quasi-isomorphism of simplicial almost A-algebras P → s.B. The components of P are free almost A-algebras, which therefore, by lemma 2.3.11, satisfy condition (L) whenever (S2) holds. In this case P itself satisfies (L).

Definition 2.4.1. The almost cotangent complex of B over A is the simplicial almost B-module

𝕃_{B/A} = B ⊗_{P} Ω_{P/A}.

For any morphism C → D of rings, let 𝕃_{D/C} be the cotangent complex defined by Illusie. Then obviously 𝕃_{B/A} = 𝕃_{B_{∗}/A_{∗}}^{a}. It is clear that Ω_{P/A} is a dimension-wise free almost P -module, in particular it is flat. The augmentation p induces an augmentation

𝕃_{B/A} → Ω_{B/A}

and by paraphrasing the argument in [4] we obtain the following

Proposition 2.4.2. (cp. [4] (II.1.2.4.2)) The natural morphism

H_{0}(𝕃_{B/A}) → Ω_{B/A}

is an isomorphism.

Corollary 2.4.3. Let M be an almost B-module. There exists a natural isomorphism

𝔼xt_{B}^{0}(𝕃_{B/A}, M) ≃ Der_{A}(B, M).

Finally we can return to our chief preoccupation, which is to extend proposition 2.3.10. First we would like to extend the homomorphism diff. This is achieved as follows. Let X be an Aextension of B by an almost B-module M. We deduce an A-extension X ∗p of P by M, hence, assuming (S2), an extension of P -modules diff(X ∗ p) of Ω_{P/A} by M and finally an element

α(X) = χ ∘ diff(X ∗ p) ∈ 𝔼xt_{P}^{1} (Ω_{P/A}, M).

On the other hand, as Ω_{P/A} is flat, [4] (I.3.3.4.4) yields natural isomorphisms

𝔼xt_{P}^{n} (Ω_{P/A}, M) ≃ 𝔼xt_{B}^{n}(𝕃_{B/A}, M). (2.4.4)

Theorem 2.4.5. (cp. [4] (III.1.2.3) Assume (S2). Then the natural homomorphism

Exal_{A}(B, M) → 𝔼xt_{B}^{1}(𝕃_{B/A}, M)

obtained by composing α and the isomorphism (2.4.4)is an isomorphism.

Proof. The proof is just the transcription of Illusie’s argument. Therefore we just outline how to construct an inverse for α and leave it at that. Let y ∈ 𝔼xt_{P}^{1} (Ω_{P/A}, M). According to proposition 2.3.16(i) there exists a quasi-isomorphism of P -modules s : M → N and an extension Y of Ω_{P/A} by N such that y = (Σs)^{−1} ∘ χ(Y). Thus we obtain an A-extension alg(Y) of P by N. As P is acyclic in degre > 0, we deduce, by applying H_{0} termwise, an A-extension H_{0}(alg(Y)) of H_{0}(P) by H_{0}(N), and finally an A-extension

β(y) = H_{0}(s)^{−1} ∗ H_{0}(alg(Y)) ∗ H_{0}(p)^{−1} ∈ Exal_{A}(B, M).

One shows using proposition 2.3.16(ii) that the result is independent of the choices of s and Y. An explicit calculation using proposition 2.3.10 then shows that β is an inverse for α.

We proceed now to list the other main properties of the almost cotangent complex.

Theorem 2.4.6. Let A → B → C be a sequence of morphisms of almost V -algebras. There exists a natural distinguished triangle of D_{•}(C)

\xymatrix{C ⊗_{B} 𝕃_{B/A} \ar[r]^{u} & 𝕃_{C/A} \ar[r]^{v} & 𝕃_{C/B} \ar[r] & C ⊗_{B} Σ𝕃_{B/A}}

where the morphisms u and v are obtained by functoriality of 𝕃.

Proof. It follows directly from [4] (II.2.1.2).

Finally we have a fundamental spectral sequence as in [4] (III.3.3.2). It goes as follows. For every integer q ≥ 0 let Sym_{B}^{q} : B − al.Mod → B − al.Mod be the non-additive functor which sends an almost B-module M to its q-th symmetric tensor power. We can extend Sym_{B}^{q} to a functor s.B − al.Mod → s.B − al.Mod by applying it termwise to the components of a simplicial B-module.

Theorem 2.4.7. (cp.[4] (III.3.3.2)) Let ϕ : R → S be a morphism of V -algebras such that S ⊗_{R} S ≃ S (e.g. such that S is a quotient of R). Let A = R^{a} and B = S^{a}. Then there is a first quadrant homology spectral sequence of bigraded almost algebras

E_{pq}^{2} = H_{p+q}(Sym_{B}^{q}(𝕃_{S/R}^{a})) ⇒ Tor_{p+q}^{A}(B, B).

Corollary 2.4.8. Under the assumptions of theorem 2.4.7 there is a five term exact sequence

Tor_{3}^{A}(B, B) → H_{3}(𝕃_{S/R}^{a}) → Λ_{B}^{2}H_{1}(𝕃_{S/R}^{a}) → Tor_{2}^{A}(B, B) → H_{2}(𝕃_{S/R}^{a}) → 0.

3. Almost ring theory

3.1. Flat, unramified andétale morphisms. Let A → B be a morphism of almost V -algebras. Using the natural “multiplication” morphism of almost A-algebras μ_{B/A} : B ⊗_{A} B → B we can see B as an almost B ⊗_{A} B-algebra.

Definition 3.1.1. Let ϕ : A → B be a morphism of almost V -algebras.

i) We say that ϕ is a flat (resp. almost projective) morphism if B is a flat (resp. almost projective) almost A-module.

ii) We say that ϕ is faithfully flat if it is flat and, for any A-module M, the natural morphism of almost A-modules ϕ ⊗_{A} 1_{M} : A ⊗_{A} M → B ⊗_{A} M is a monomorphism.

iii) We say that ϕ is almost finite if B is an almost finitely generated almost A-module.

iv) We say that ϕ is unramified if B is an almost projective almost B ⊗_{A} B-module (via the morphism μ_{B/A} defined above).

v) We say that ϕ is étale if it is flat and unramified.

Lemma 3.1.2. (see [3]) Let ϕ : A → B and ψ : B → C be morphisms of almost V -algebras.

i) Any base change of a flat (resp. almost projective, resp. faithfully flat, resp. almost finite, resp. unramified, resp. étale) morphism is flat (resp. almost projective, resp. faithfully flat, resp. almost finite, resp. unramified, resp. étale);

ii) if both ϕ and ψ are flat (resp. almost projective, resp. faithfully flat, resp. almost finite, resp. unramified, resp. étale), then so is ψ ∘ ϕ;

iii) if ϕ is flat and ψ ∘ ϕ is faithfully flat, then ϕ is faithfully flat;

iv) if ϕ is unramified and ψ ∘ ϕ is flat (resp. étale), then ψ is flat (resp. étale).

Recall the topological meaning of idempotents: if A is a ring and e ∈ A satisfies the relation e^{2} = e, then the ideal eA (resp. (1 − e)A) is also a ring with the identity given by e (resp. by 1 − e). Then the natural A-linear morphism eA ⊕ (1 − e)A → A is in fact an isomorphism of rings, so that Spec(A) decomposes as the union of two disjoint open and closed subspaces.

In almost ring theory we will find useful to study certain “approximate idempotents”, as in the following proposition, whose proof can be found in [3].

Proposition 3.1.3. A morphism ϕ : A → B is unramified if and only if there exists an almost A-element e_{B/A} ∈ B ⊗_{A} B_{∗} such that

i) e_{B/A}^{2} = e_{B/A};

ii) μ_{B/A}(e_{B/A}) = \underline{1};

iii) x · e_{B/A} = 0 for all x ∈ I_{B/A∗}.

Corollary 3.1.4. Under the hypotheses and notation of the proposition the ideal I = I_{B/A} has a natural structure of almost A-algebra, with unit morphism given by \underline{1}_{I/A} = \underline{1}_{B⊗_{A}B/A} − e_{B/A} and whose multiplication is the restriction of μ_{B⊗_{A}B/A} to I. Moreover the natural morphism

B ⊗_{A} B → I_{B/A} ⊕ B x ↦ (x · \underline{1}_{I/A} ⊕ μ_{B/A}(x))

is an isomorphism of almost A-algebras.

Proof. Left to the reader as an exercise.

In order to manipulate idempotents we will need the following almost version of a well known lifting trick.

Lemma 3.1.5. Assume (S1) and let ϕ : A → B be an epimorphism of almost V -algebras such that I = Ker(ϕ) is a nilpotent ideal, so that I^{m} = 0 for some positive integer m. Suppose that \overline{e} is an idempotent almost element of B. Then there exists a unique lifting e ∈ ϕ^{−1}(\overline{e}) which is also idempotent.

Proof. Suppose first that I^{2} = 0. Pick any ε ∈ 𝔪 and choose an element x ∈ ϕ^{−1}(ε · \overline{e}). Let y = x − ε · \underline{1}. Clearly x · y ∈ I. We write ε^{3} · \underline{1} = (x + y)^{3} = e_{ε} + e′_{ε} where

\begin{matrix}e_{ε} = & x^{3} + 3 · x^{2} · y \\ e′_{ε} = & 3 · x · y^{2} + y^{3}. \end{matrix}

It is easy to verify that e_{ε} · (\underline{1} − e_{ε}) = e_{ε} · e′_{ε} = 0, that is e_{ε}^{2} = ε^{3} · e_{ε}. Moreover ϕ(e_{ε}) = ε^{3} · \overline{e}. If δ ∈ 𝔪 is any other element, we have

0 = (δ^{3} · e_{ε} − ε^{3} · e_{δ})^{3} = δ^{9} · e_{ε}^{3} − ε^{9} · e_{δ}^{3} + 3δ^{3} · e_{ε} · ε^{6} · e_{δ}^{2} − 3δ^{6} · e_{ε}^{2} · ε^{3} · e_{δ}

whence

δ^{9} · e_{ε}^{3} = ε^{9} · e_{δ}^{3}. (3.1.6)

Let us define a morphism e : 𝔪 ⊗_{V} 𝔪 → A_{∗} as follows. For x, y ∈ 𝔪 choose m_{i} ∈ V and s_{i} ∈ S such that y = ∑_{i} m_{i} · s_{i}^{9} (under (S1) these can always be found). Then we set

x ⊗ y ↦ x · ∑_{i} m_{i} · e_{s_{i}}^{3}.

To check that e is well-defined, pick also a similar decomposition x = ∑_{j} r_{j} · q_{j}^{9} with r_{j} ∈ V, q_{j} ∈ S. We derive from (3.1.6)

x · ∑_{i} m_{i} · e_{s_{i}}^{3} = ∑_{ij} r_{j} · q_{j}^{9} · m_{i} · e_{s_{i}}^{3} = ∑_{ij} r_{j} · m_{i} · s_{i}^{9} · e_{t_{j}}^{3} = ∑_{j} r_{j} · y · e_{q_{j}}^{3}

which shows that e(x ⊗ y) does not depend on the choice of decomposition for y. Similarly one proves that the map is bilinear in x and y. Then e defines an almost element of A such that ϕ(e) = \overline{e} and the same kind of arguments using (3.1.6) shows that e is the unique idempotent of A_{∗} with this property.

For the general case I^{m} = 0, one proceeds by induction, first lifting to A/I^{m−1} and then to A.

Theorem 3.1.7. Assume (S1) and let ϕ : A → B be an étale morphism of almost algebras. Then Exal_{A}(B, M) = 0 for all almost B-modules M.

Proof. Let X = (0 → M →^{ι} C →^{α} B → 0) be an A-extension of B by the almost B-module M. We need to construct a splitting of almost A-algebras s : B → C. As ϕ is flat, the diagram

B ⊗_{A} X = (0 → B ⊗_{A} M → B ⊗_{A} C → B ⊗_{A} B → 0)

is an A-extension of B ⊗_{A} B. Let σ_{M} : B ⊗_{A} M → M be the scalar multiplication morphism for the almost B-module M and set

Y = σ_{M} ∗ (B ⊗_{A} X) = (0 → M →^{ι′} D →^{α′} B ⊗_{A} B → 0).

On B ⊗_{A} M we have also the structure of almost B ⊗_{A} B-module mandated by (2.1.6). Then we have a well-defined structure of almost B ⊗_{A} B-module on M, characterized by requiring the morphism σ_{M} to be B ⊗_{A} B-linear. Let σ′_{M} : (B ⊗_{A} B) ⊗_{A} M → M be the corresponding scalar product; in conformity with (2.1.4)we have the equality :

σ′_{M} ∘ (1_{B⊗_{A}B} ⊗_{A} σ_{M}) = σ_{M} ∘ σ_{B⊗_{A}M}. (3.1.8)

Furthermore, let e ∈ B ⊗_{A} B_{∗} be the idempotent almost element provided by proposition 3.1.3. We define a B-linear morphism j : B → B ⊗_{A} B by b ↦ e · (1 ⊗ b) for all b ∈ B_{∗}. Notice that j is in fact a morphism of non-unital almost V -algebras. Form the diagram Y ∗ j = (0 → M →^{ι′′} E →^{α′′} B → 0); this is an A-extension of non-unital almost V -algebras. We construct a splitting B → E as follows. Let f ∈ D_{∗} be the unique lifting of the idempotent e provided by lemma 3.1.5 and denote by π : B ⊗_{A} C → D the projection induced by σ_{M}. Define a morphism of non-unital almost V -algebras ψ_{B} : B → D by b ↦ f · π(b ⊗ 1) for all b ∈ B_{∗}. Since α′ ∘ ψ_{B} = j, the pair (ψ_{B}, 1_{B}) induces a morphism of non-unital almost V -algebras ω : B → E which splits α′′. Notice that ω(1) = f. We want to show that Y ∗ j is isomorphic to X in the category of A-extensions of non-unital almost V -algebras. To this purpose we need to construct a morphism of non-unital almost V -algebras β : C → E such that α′′ ∘ β = α and β ∘ ι = ι′′. This is achieved as follows. Define a morphism ψ_{C} : C → D of non-unital almost V -algebras by c ↦ f · π(1 ⊗ c) (c ∈ C_{∗}). We verify that α′ ∘ ψ_{C} = j ∘ α. Indeed

α′ ∘ ψ_{C}(c) = α′(f · π(1 ⊗ c)) = e · (1 ⊗ α(c)) = e · (α(c) ⊗ 1) = j ∘ α(c).

Therefore the pair (ψ_{C}, α) determines a morphism β : C → E of non-unital almost V -algebras such that α′′ ∘ β = α. It remains to verify that β ∘ ι = ι′′. It suffices to show that ψ_{C} ∘ ι = ι′. However we have ψ_{C} ∘ ι(m) = f · π(1 ⊗ m) = e · σ_{M}(1 ⊗ m) because M is an almost B ⊗_{A} Bmodule with the scalar multiplication σ′_{M} introduced above. Then we compute using (3.1.8), (2.1.3)and (2.1.6):

\begin{matrix}e · σ_{M}(1 ⊗ m) = & σ′_{M}(e ⊗ σ_{M}(1 ⊗ m)) = σ_{M} ∘ σ_{B⊗_{A}M}(e ⊗ 1 ⊗ m) \\ = & σ_{M} ∘ (μ_{B/A} ⊗ σ_{M}) ∘ (1_{B} ⊗ η_{B|B} ⊗ 1_{M})(e ⊗ 1 ⊗ m) \\ = & σ_{M} ∘ (μ_{B/A} ⊗ 1_{M}) ∘ (1_{B⊗_{A}B} ⊗ σ_{M})((1_{B} ⊗ η_{B|B})(e ⊗ 1) ⊗ m) \\ = & σ_{M} ∘ (1_{B} ⊗ σ_{M}) ∘ (1_{B⊗_{A}B} ⊗ σ_{M})((1_{B} ⊗ η_{B|B})(e ⊗ 1) ⊗ m) \\ = & σ_{M} ∘ (1_{B} ⊗ (σ_{M} ∘ (1_{B} ⊗ σ_{M})))((1_{B} ⊗ η_{B|B})(e ⊗ 1) ⊗ m) \\ = & σ_{M} ∘ (1_{B} ⊗ (σ_{M} ∘ (μ_{B/A} ⊗ 1_{M})))((1_{B} ⊗ η_{B|B})(e ⊗ 1) ⊗ m) \\ = & σ_{M} ∘ (1_{B} ⊗ σ_{M})((1_{B} ⊗ (μ_{B/A} ∘ η_{B|B}))(e ⊗ 1) ⊗ m) \\ = & σ_{M} ∘ (1_{B} ⊗ σ_{M})((1_{B} ⊗ μ_{B/A})(e ⊗ 1) ⊗ m) \\ = & σ_{M} ∘ (μ_{B/A} ⊗ 1_{M})((1_{B} ⊗ μ_{B/A})(e ⊗ 1) ⊗ m) \\ = & σ_{M}((μ_{B/A} ∘ (1_{B} ⊗ μ_{B/A}))(e ⊗ 1) ⊗ m) \\ = & σ_{M}((μ_{B/A} ∘ (μ_{B/A} ⊗ 1_{B}))(e ⊗ 1) ⊗ m) \\ = & σ_{M}(μ_{B/A}(1 ⊗ 1) ⊗ m) = σ_{M}(1 ⊗ m) \\ = & m. \end{matrix}

It follows in particular that β is an isomorphism of non-unital almost algebras. Then the morphism s = β^{−1} ∘ω : B → C is a splitting for α in the category of non-unital almost V -algebras. But we remark that β(1) = f which means s(1) = 1 so that s is actually a homomorphism of (unital) almost V -algebras. The theorem follows.

3.2. Lifting theorems.

Lemma 3.2.1. (see [7]) Let A → B be an epimorphism of almost V -algebras with kernel I. Let U be the A-extension 0 → I/I^{2} → A/I^{2} → B → 0. Then the assignment f ↦ f ∗ U defines a natural isomorphism

\xymatrix{Hom_{B}(I/I^{2}, M) \ar[r]^{∼} & Exal_{A}(B, M).} (3.2.2)

Now consider any morphism of A-extensions

\xymatrix{B̃ : \ar[d]^{f̃} & 0 \ar[r] & I \ar[r] \ar[d]^{u} & B \ar[r] \ar[d]^{f} & B_{0} \ar[r] \ar[d]^{f_{0}} & 0 \\ C̃ : & 0 \ar[r] & J \ar[r] & C \ar[r] & C_{0} \ar[r] & 0.} (3.2.3)

The morphism u induces by adjunction a morphism of almost C_{0}-modules

C_{0} ⊗_{B_{0}} I → J (3.2.4)

whose image is the ideal I · C, so that the square diagram of almost algebras defined by f̃ is cofibred (i.e. C_{0} ≃ C ⊗_{B} B_{0}) if and only if (3.2.4)is an epimorphism.

Lemma 3.2.5. Let f̃: B̃ → C̃ be a morphism of A-extensions as above, such that the corresponding square diagram of almost algebras is cofibred. Then the morphism f : B → C is flat if and only if f_{0} : B_{0} → C_{0} is flat and (3.2.4)is an isomorphism.

Proof. It follows directly from the (almost version of the) local flatness criterion (see [6] Th. 22.3).

We are now ready to put together all the work done so far and begin the study of deformations of almost algebras. From here on, throughout the rest of this section we assume that axiom (S2) is verified.

The morphism u : I → J is an element in Hom_{B_{0}}(I, J); by lemma 3.2.1 the latter group is naturally isomorphic to Exal_{B}(B_{0}, J). By applying transitivity (theorem 2.4.6) to the sequence of morphisms B → B_{0} →^{f_{0}} C_{0} we obtain an exact sequence of abelian groups

Exal_{B_{0}}(C_{0}, J) → Exal_{B}(C_{0}, J) → Hom_{B_{0}}(I, J) →^{∂} 𝔼xt_{C_{0}}^{2}(𝕃_{C_{0}/B_{0}}, J).

Hence we can form the element

ω(B̃, f_{0}, u) = ∂(u) ∈ 𝔼xt_{C_{0}}^{2}(𝕃_{C_{0}/B_{0}}, J).

The proof of the next result goes exactly as in [4] (III.2.1.2.3).

Proposition 3.2.6. i) Let the A-extension B̃, the B_{0}-linear morphism u : I → J and the morphism of almost A-algebras f_{0} : B_{0} → C_{0} be given as above. Then there exists an Aextension C̃ and a morphism f̃: B̃ → C̃ completing diagram (3.2.3)if and only if ω(B̃, f_{0}, u) = 0. (i.e. ω(B̃, f_{0}, u) is the obstruction to the lifting of B̃ over f_{0}.)

ii) Assume that the obstruction ω(B̃, f_{0}, u) vanishes. Then the set of isomorphism classes of A-extensions C̃ as in (i) forms a torsor under the group Exal_{B_{0}}(C_{0}, J) (≃ 𝔼xt_{C_{0}}^{1}(𝕃_{C_{0}/B_{0}}, J)).

iii) The group of automorphisms of an A-extension C̃ as in (i) is naturally isomorphic to Der_{B_{0}}(C_{0}, J) (≃ 𝔼xt_{C_{0}}^{0}(𝕃_{C_{0}/B_{0}}, J)).

The obstruction ω(B̃, f_{0}, u) depends functorially on u. More exactly, if we denote by

ω(B̃, f_{0}) ∈ 𝔼xt_{C_{0}}^{2}(𝕃_{C_{0}/B_{0}}, C_{0} ⊗_{B_{0}} I)

the obstruction corresponding to the natural morphism I → C_{0} ⊗_{B_{0}} I, then for any other morphism u : I → J we have

ω(B̃, f_{0}, u) = v ∘ ω(B̃, f_{0})

where v is the morphism (3.2.4). Taking lemma 3.2.5 into account we deduce

Corollary 3.2.7. Suppose that B_{0} → C_{0} is flat. Then

(i) The class ω(B̃, f_{0}) is the obstruction to the existence of a flat deformation of C_{0} over B, i.e. of a B-extension C̃ as in (3.2.3) such that C is flat over B and C ⊗_{B} B_{0} → C_{0} is an isomorphism.

(ii) When the obstruction ω(B̃, f_{0}) vanishes, the set of isomorphism classes of flat deformations of C_{0} over B forms a torsor under the group Exal_{B_{0}}(C_{0}, C_{0} ⊗_{B_{0}} I).

(iii) The group of automorphisms of a given flat deformation of C_{0} over B is naturally isomorphic to Der_{B_{0}}(C_{0}, C_{0} ⊗_{B_{0}} I).

Now, suppose we are given two A-extensions C̃^{1}, C̃^{2} with morphisms of A-extensions

\xymatrix{B̃ : \ar[d]^{f̃^{i}} & 0 \ar[r] & I \ar[r] \ar[d]^{u^{i}} & B \ar[r] \ar[d]^{f^{i}} & B_{0} \ar[r] \ar[d]^{f_{0}^{i}} & 0 \\ C̃^{i} : & 0 \ar[r] & J^{i} \ar[r] & C^{i} \ar[r] & C_{0}^{i} \ar[r] & 0}

and morphisms v : J^{1} → J^{2}, g_{0} : C_{0}^{1} → C_{0}^{2} such that

u^{2} = v ∘ u^{1} and f_{0}^{2} = g_{0} ∘ f_{0}^{1}. (3.2.8)

We consider the problem of finding a morphism of A-extensions

\xymatrix{C̃^{1} : \ar[d]^{g̃} & 0 \ar[r] & J^{1} \ar[r] \ar[d]^{v} & C^{1} \ar[r] \ar[d]^{g} & C_{0}^{1} \ar[r] \ar[d]^{g_{0}} & 0 \\ C̃^{2} : & 0 \ar[r] & J^{2} \ar[r] & C^{2} \ar[r] & C_{0}^{2} \ar[r] & 0} (3.2.9)

such that f̃^{2} = g̃ ∘ f̃^{1}. Let us denote by e(C̃^{i}) ∈ 𝔼xt_{C_{0}^{i}}^{1}(𝕃_{C_{0}^{i}/B}, J^{i}) the classes defined by the

B-extensions C̃^{1}, C̃^{2} via the isomorphism of theorem 2.4.5 and by

v∗ : 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}, J^{1}) → 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}, J^{2})
∗g_{0} : 𝔼xt_{C_{0}^{2}}^{1}(𝕃_{C_{0}^{2}/B}, J^{2}) → 𝔼xt_{C_{0}^{2}}^{1}(C_{0}^{2} ⊗_{C_{0}^{1}} 𝕃_{C_{0}^{1}/B}, J^{2})

the canonical morphisms defined by v and g_{0}. Using the natural isomorphism

𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}, J^{2}) ≃ 𝔼xt_{C_{0}^{2}}^{1}(C_{0}^{2} ⊗_{C_{0}^{1}} 𝕃_{C_{0}^{1}/B}, J^{2})

we can identify the target of both v∗ and ∗g with 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}, J^{2}). It is clear that the problem admits a solution if and only if the A-extensions v ∗ C̃^{1} and C^{2} ∗ g_{0} coincide, i.e. if and only if v ∗ e(C^{1}) − e(C^{2}) ∗ g_{0} = 0. By applying transitivity to the sequence of morphisms B → B_{0} → C_{0}^{1} we obtain an exact sequence

0 → 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B_{0}}, J^{2}) → 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}, J^{2}) → Hom_{C_{0}^{1}}(C_{0}^{1} ⊗_{B_{0}} I, J^{2})

It follows from (3.2.8)that the image of v∗e(C^{1})−e(C^{2})∗g_{0} in the group Hom_{C_{0}^{1}}(C_{0}^{1}⊗_{B_{0}} I, J^{2}) vanishes, therefore

v ∗ e(C^{1}) − e(C^{2}) ∗ g_{0} ∈ 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B_{0}}, J^{2}). (3.2.10)

In conclusion, we derive the following result as in [4] (III.2.2.2).

Proposition 3.2.11. With the above notations, the class (3.2.10) is the obstruction to the existence of a morphism of A-extensions g̃: C̃^{1} → C̃^{2} as in (3.2.9)such that f̃^{2} = g̃∘ f̃^{1}. When the obstruction vanishes, the set of such morphisms forms a torsor under the group Der_{B_{0}}(C_{0}^{1}, J^{2}) (the latter being identified with 𝔼xt_{C_{0}^{2}}^{0}(C_{0}^{2} ⊗_{C_{0}^{1}} 𝕃_{C_{0}^{1}/B_{0}}, J^{2})).

Let B be an almost A-algebra. We can find a set S and an epimorphism of almost A-algebras ψ : A[S] → B. Let J = Ker(ψ). There is a natural isomorphism Tor_{1}^{A[S]}(B, B) ≃ J/J^{2} which extends to a natural homomorphism of graded anticommutative almost B-algebras

⊕_{n≥0}Λ_{B}^{n}(J/J^{2}) → ⊕_{n≥0}Tor_{n}^{A[S]}(B, B) (3.2.12)

(for the natural product structures on the exterior algebra and the almost Tor-algebra).

Proposition 3.2.13. Keep the above hypotheses and suppose moreover that B is anétale almost A-algebra. Then J/J^{2} is a free almost B-module and the morphism (3.2.12)is an isomorphism of graded anticommutative almost algebras.

Proof. By hypothesis B is a flat almost A-algebra, hence B[S] is a flat almost A[S]-algebra. Moreover the morphism ψ factors as a composition

\xymatrix{A[S] \ar[r] & B[S] \ar[r]^{1_{B}⊗_{A}ψ} & B ⊗_{A} B \ar[r]^{μ_{B/A}} & B}

thereby inducing on B a structure of almost B[S]-module which restricts to the given almost A[S]-module structure. Hence we deduce, by flat base change (see [8] (3.2.9)), a natural isomorphism of graded almost algebras

⊕_{n≥0}Tor_{n}^{A[S]}(B, B) ≃ ⊕_{n≥0}Tor_{n}^{B[S]}(B ⊗_{A} B, B).

On the other hand corollary 3.1.4 implies

Tor_{n}^{B[S]}(B ⊗_{A} B, B) ≃ Tor_{n}^{B[S]}(B, B) ⊕ Tor_{n}^{B[S]}(I_{B/A}, B)

for all integers n. The morphism η = μ_{B/A} ∘ (1_{B} ⊗_{A} ψ) : B[S] → B is determined by a map S → B_{∗} : s ↦ b_{s}. Define a B-automorphism ϕ : B[S] → B[S] by s ↦ s − b_{s}. We derive a commutative diagram with short exact rows

[figure: 0; S · B[S]; ≃; B[S]; ϕ; B; 0; 0; J_{1} = Ker(η); B[S]; η; B; 0] .

If we denote by π : F = B[S]^{(S)} → J_{1} the epimorphism coming from the above map ϕ : S · B[S] → J_{1}, then the Koszul complex K_{•}(π) associated to π is a flat resolution of B as a B[S]-module. Recall that K_{n}(π) = Λ_{B[S]}^{n}(F) and the differential of the complex is given by the formula

d_{n} : Λ_{B[S]}^{n+1}(F) → Λ_{B[S]}^{n}(F)
x_{0} ∧ ... ∧ x_{n} ↦ ∑_{j=0}^{n}(−1)^{j}π(x_{j}) · (x_{0} ∧ ... ∧ x_{j−1} ∧ x_{j+1} ∧ ... ∧ x_{n}).

We claim that Tor_{n}^{B[S]}(I_{B/A}, B) vanishes for all integers n. It suffices to show that the complex K_{•}(π) ⊗_{B[S]} I_{B/A} is acyclic. Let e be the idempotent of B ⊗_{A} B provided by proposition 3.1.3 and for any ε ∈ S choose f_{ε} ∈ B_{∗}[S] such that (1_{B} ⊗_{A} ψ)(f_{ε}) = ε · (1 − e). Clearly f_{ε} ∈ J_{1}_{∗}, hence we can find g_{ε} ∈ F_{∗} with π(g_{ε}) = f_{ε} and we can define a sequence of morphisms

s_{n} : Λ_{B[S]}^{n}(F) → Λ_{B[S]}^{n+1}(F) x_{0} ∧ ... ∧ x_{n−1} ↦ g_{ε} ∧ x_{0} ∧ ... ∧ x_{n−1}.

A straightforward computation yields

(d_{n} ∘ s_{n} + s_{n−1} ∘ d_{n−1}) ⊗_{B[S]} 1_{I_{B/A}} = ε · 1_{K_{n}(π)⊗_{B[S]}I_{B/A}}.

As ε can be chosen arbitrarily small, the claim follows. Summing up we get an isomorphism of graded almost algebras

⊕_{n≥0}Tor_{n}^{A[S]}(B, B) ≃ ⊕_{n≥0}Tor_{n}^{B[S]}(B, B).

To determine the right-hand side we use again the Koszul complex. The claim follows.

Proposition 3.2.14. Suppose that B is an étale almost A-algebra. Then we have

H_{q}(𝕃_{B/A}) ≃ 0 0 ≤ q ≤ 2.

Proof. For q = 0 the claim follows easily from corollary 2.4.3 and corollary 3.1.4. For q = 1 the claim follows from the standard cohomology spectral sequence

E_{2}^{pq} = Ext_{B}^{p}(H_{q}(𝕃_{B/A}), M) ⇒ 𝔼xt_{B}^{p+q}(𝕃_{B/A}, M) (3.2.15)

together with theorem 2.4.5 and theorem 3.1.7. Finally, pick a set S large enough and a morphism A[S] → B of almost A-algebras such that A_{∗}[S] → B_{∗} is surjective. Applying transitivity (theorem 2.4.6) to the sequence A → A[S] → B we see easily that H_{p}(𝕃_{B/A}) ≃ H_{p}(𝕃_{B_{∗}/A_{∗}[S]}^{a}) for all p ≥ 2. Then proposition 3.2.13 and corollary 2.4.8 yield H_{2}(𝕃_{B/A}) ≃ 0.

For a given almost V -algebra A, let Ét(A) be the category of étale almost A-algebras. Notice that, by lemma 3.1.2(iv) all morphisms in Ét(A) are étale.

Theorem 3.2.16. i) Let A → B be an étale morphism of almost algebras. Let C be any almost A-algebra and I ⊂ C a nilpotent ideal. Then the natural morphism

Hom_{A−al.Alg}(B, C) → Hom_{A−al.Alg}(B, C/I)

is bijective.

ii) Let A be an almost V -algebra and I ⊂ A a nilpotent almost ideal. Set A′ = A/I. Then the natural functor

Ét(A) → Ét(A′) (ϕ : A → B) ↦ (ϕ′ = 1_{A′} ⊗_{A} ϕ : A′ → A′ ⊗_{A} B)

is an equivalence of categories.

Proof. By induction we can assume that I^{2} ≃ 0. Then, taking into account proposition 3.2.14 and the spectral sequence (3.2.15), claim (i) follows directly from proposition 3.2.11.

We show (ii) : by corollary 3.2.7 (and again proposition 3.2.14) we know that we can lift an étale morphism ϕ′ : A′ → B′ to a flat morphism ϕ : A → B (such that B′ ≃ A′ ⊗_{A} B). We need to verify that ϕ is étale. To this purpose we apply proposition 3.1.3 : since ϕ′ is étale, there is an idempotent e′ = e_{B′/A′} ∈ B′ ⊗_{A′} B′_{∗} such that μ_{B′/A′}(e′) = \underline{1} and x · e′ = 0 for all x ∈ I_{B′/A′∗}. By lemma 3.1.5 there exists a unique lifting of e′ to an idempotent e ∈ B ⊗_{A} B_{∗}. Clearly μ_{B/A}(e) ∈ B_{∗} is a lifting of the idempotent μ_{B′/A′}(e′) = \underline{1} ∈ B′_{∗}, hence μ_{B/A}(e) = \underline{1} by lemma 3.1.5(iii). It remains to verify that x·e = 0 for all x ∈ I_{B/A∗}. We consider the morphism

∂ : B → B ⊗_{A} B b ↦ (\underline{1} ⊗ b − b ⊗ \underline{1}) · e.

It is easily checked that the image of ∂ is contained in the ideal I(B ⊗_{A} B). Moreover ∂ factors through the projection B → B′ and we can denote

∂′ : B′ → I(B ⊗_{A} B)

the induced morphism. Let J be the B⊗_{A}B-submodule generated by the image of ∂. We remark that the annihilator of J contains I_{B/A} = Ker(μ_{B/A}) (so that the B ⊗_{A} B-module structure on J is obtained by restriction of scalars via μ_{B/A}). In fact we have, for all b, b′ ∈ B_{∗}

(\underline{1} ⊗ b − b ⊗ \underline{1}) · e · (\underline{1} ⊗ b′ − b′ ⊗ \underline{1}) = \{(\underline{1} ⊗ b − b ⊗ \underline{1}) · e\} · \{e · (\underline{1} ⊗ b′ − b′ ⊗ \underline{1})\} = 0

because I^{2} = 0. Pick b_{1}, b_{2} ∈ B_{∗}. We compute

∂(b_{1}b_{2}) = ∂(b_{1}) · (\underline{1} ⊗ b_{2}) + (b_{1} ⊗ \underline{1}) · ∂(b_{2}).

In other words, ∂′ is an almost A-derivation of B′ with values in the almost B′-module J. Now it follows from corollary 2.4.3 and proposition 3.2.14 that ∂′ = 0 and therefore ∂ = 0. This proves that ϕ is unramified, hence étale as claimed.

Acknowledgements I am very thankful to Ofer Gabber for many useful discussions and for pointing out some mistakes in the first version of this article.

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[3] O.Gabber and L.Ramero, Almost ring theory - second release. Preprint Lab. Math. Pures de Bordeaux 105 (1999).

[4] L.Illusie, Complexe cotangent et déformations I. Lect. Notes Math. 239; Berlin Heidelberg New York: Springer (1971).

[5] S.Mac Lane, Categories for the working mathematician. Grad. Text Math. 5; Berlin Heidelberg New York: Springer (1971).

[6] H.Matsumura, Commutative ring theory. Cambridge Univ. Press (1980).

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Lorenzo Ramero Université de Bordeaux I Laboratoire de Mathématiques Pures 351, cours de la Liberation F-33405 Talence Cedex

E-mail address: ramero@math.u-bordeaux.fr