postscript version (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.
postscript version 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 p-adic Hodge theory.
This sixth release has been uploaded on July 18 (2002).
Updated December 7, 2005.
In this article I revisit my old theme of the
study of the topology of the p-adic punctured disc
(see the above "On a class of étale analytic sheaves").
It benefits of the new techniques developed by Roland Huber.
This is the final release, reformatted using Springer's
TeX macros. It will appear in the volume 102 of the
Publications Mathématiques de l'IHÉS.
pdf version 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
postscript version 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.