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The p-adic Simpson Correspondence (AM-193)$
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Ahmed Abbes, Michel Gros, and Takeshi Tsuji

Print publication date: 2016

Print ISBN-13: 9780691170282

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691170282.001.0001

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Almost étale coverings

Almost étale coverings

Chapter:
(p.449) Chapter V Almost étale coverings
Source:
The p-adic Simpson Correspondence (AM-193)
Author(s):

Ahmed Abbes

Michel Gros

Takeshi Tsuji

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691170282.003.0005

This chapter explains Faltings' theory of almost étale extensions, a tool that has become essential in many questions in arithmetic geometry, even beyond p-adic Hodge theory. It begins with a brief historical overview of almost étale extensions, noting how Faltings developed the “almost purity theorem” and proved the Hodge–Tate decomposition of the étale cohomology of a proper smooth variety. The chapter proceeds by discussing almost isomorphisms, almost finitely generated projective modules, trace, rank and determinant, almost flat modules and almost faithfully flat modules, almost étale coverings, almost faithfully flat descent, and liftings. Finally, it describes group cohomology of discrete A–G-modules and Galois cohomology.

Keywords:   almost étale extension, étale cohomology, almost isomorphism, almost flat module, almost faithfully flat module, almost étale covering, almost faithfully flat descent, discrete A–G-module, Galois cohomology

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