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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. An overview

Representations of the fundamental group and the torsor of deformations. An overview

(p.1) Chapter I Representations of the fundamental group and the torsor of deformations. An overview
The p-adic Simpson Correspondence (AM-193)

Ahmed Abbes

Michel Gros

Princeton University Press

This chapter provides an overview of a new approach to the p-adic Simpson correspondence, focusing on representations of the fundamental group and the torsor of deformations. The discussion covers the notation and conventions, small generalized representations, the torsor of deformations, Faltings ringed topos, and Dolbeault modules. The chapter begins with a short aside on small generalized representations in the affine case, which will be used as intermediary for the study of Dolbeault representations. It then introduces the notion of generalized Dolbeault representation for a small affine scheme and the companion notion of solvable Higgs module, and constructs a natural equivalence between these two categories. It establishes links between these notions and Faltings smallness conditions and relates this to Hyodo's theory. It also describes the Higgs–Tate algebras and concludes with an analysis of the logical links for a Higgs bundle, between smallness and solvability.

Keywords:   p-adic Simpson correspondence, torsor, deformation, small generalized representation, Faltings ringed topos, Dolbeault module, Dolbeault representation, solvable Higgs module, Higgs–Tate algebra, Higgs bundle

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