- Effective estimates for unirationality (appeared in Manuscripta Mathematica , n.68 (1990);
- On a class of étale analytic sheaves postscript version
- Almost ring theory - sixth (and final) release postscript version
- Fragments of almost ring theory postscript version
- Local monodromy in non-archimedean analytic geometry - final release Updated December 7, 2005. In this article I revisit my old theme of the study of the topology of the
- Almost rings and perfectoid spaces - Release 7.5 pdf version
- Hasse-Arf filtrations in $p$-adic analytic geometry - Fourth Release postscript version

Deals with some old question of unirationality of projective varieties, going back to the "italian school" of algebraic geometry.

(appeared in Journal of Algebraic Geometry , n.3 (1998);

this evolved from my PhD thesis. It deals with the question of understanding the ramification of étale sheaves over a non-archimedean analytic curve (à la Berkovich). Subsequently, Roland Huber has found a better way of studying this question : his results are much more general and complete than mine; he also verifies that his definition of the Swan conductor agrees with mine, when they both are defined.

In collaboration with Ofer Gabber. A preliminary version of this work (authored by me alone) appeared in the IHES preprint series in 1997. Almost ring theory, a domain at the crossroads of commutative algebra and category theory, is the natural outgrowth of Faltings' method of almost étale extensions. The paper provides comprehensive foundations for Faltings' approach to

Septembre 2003 : "Almost Ring Theory" has appeared as the Volume 1800 of the Springer Lecture Notes in Mathematics.

The published version features a completely rewritten introduction and contains several small improvements; also, a large number of minor mistakes have been corrected. The cover illustration has been made for us by my colleague Michel Mendes-France.

This paper collects some material that was originally in the first version of the paper on almost ring theory, but that did not make it to the second release. It has appeared in the volume 104 of Rendiconti del Seminario Matematico dell'Università di Padova

September 28, 2018. This is Release 7.5 of our project, aiming to provide complete treatment of the foundations of almost ring theory, following and extending Faltings' method of "almost étale extensions". We have replaced Faltings's proof of the almost purity theorem with Scholze's new methods, based on his theory of perfectoid spaces. Besides the complete proof of the almost purity theorem, it includes a chapter of applications; especially, we prove a generalization of the direct summand conjecture, extending the work of Y.André

appeared in Journal of Algebraic Geometry n.21 (2012).

Updated May 10, 2009. This is the same article that was previously called "Local monodromy in non-archimedean analytic geometry -- II" : I have changed the title and modified the introduction, for marketing reasons. Moreover, I have added an application to Deligne's problem of the localization of the determinant of cohomology. Namely, I prove a factorization as tensor product of so-called cohomological epsilon-factors. An unusual feature is that these factors are semilinear Galois representations (rather than linear ones). Apart from the above, the main theme is the study of the monodromy of local systems with bounded ramification on a disc defined over a non-archimedean valued field of characteristic zero. In this fourth (and hopefully final) release, I first construct the local Fourier transforms, and establish their main properties. Also, I show that the Fourier transform of a "perverse sheaf" with bounded ramification on the affine line, is again a "perverse sheaf" with bounded ramification (I put that in quotes, because actually the language of perversity is not used). These foundations are then used to exhibit a natural break decomposition for local systems with bounded ramification, analogous to the classical one for Galois representations.