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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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Representations of the fundamental group and the torsor of deformations. Local study

Representations of the fundamental group and the torsor of deformations. Local study

Chapter:
(p.27) Chapter II Representations of the fundamental group and the torsor of deformations. Local study
Source:
The p-adic Simpson Correspondence (AM-193)
Author(s):

Ahmed Abbes

Michel Gros

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691170282.003.0002

This chapter focuses on representations of the fundamental group and the torsor of deformations. It considers the case of an affine scheme of a particular type, qualified also as small by Faltings. It introduces the notion of Dolbeault generalized representation and the companion notion of solvable Higgs module, and then constructs a natural equivalence between these two categories. It proves that this approach generalizes simultaneously Faltings' construction for small generalized representations and Hyodo's theory of p-adic variations of Hodge–Tate structures. The discussion covers the relevant notation and conventions, results on continuous cohomology of profinite groups, objects with group actions, logarithmic geometry lexicon, Faltings' almost purity theorem, Faltings extension, Galois cohomology, Fontaine p-adic infinitesimal thickenings, Higgs–Tate torsors and algebras, Dolbeault representations, and small representations. The chapter also describes the descent of small representations and applications and concludes with an analysis of Hodge–Tate representations.

Keywords:   torsor, deformation, Dolbeault generalized representation, solvable Higgs module, Hyodo's theory, Hodge–Tate structure, Faltings extension, Galois cohomology, small representation, Hodge–Tate representation

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