The p-adic Simpson Correspondence (AM-193)

by Ahmed Abbes, Michel Gros, Takeshi Tsuji

Overview

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches, one by Ahmed Abbes and Michel Gros, the other by Takeshi Tsuji. The authors mainly focus on generalized representations of the fundamental group that are p-adically close to the trivial representation.

The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The authors show the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the volume contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored. The authors present a new approach based on a generalization of P. Deligne's covanishing topos.

Product Details

ISBN-13:	9780691170282
Publisher:	Princeton University Press
Publication date:	02/09/2016
Series:	Annals of Mathematics Studies , #193
Pages:	616
Product dimensions:	7.10(w) x 10.00(h) x 1.50(d)

About the Author

Ahmed Abbes is director of research at the French National Center for Scientific Research (CNRS) and the Institute of Advanced Scientific Studies (IHÉS), France. Michel Gros is a researcher at the CNRS. Takeshi Tsuji is a professor in the Graduate School of Mathematical Sciences at the University of Tokyo.

Table of Contents

Foreword ix

Chapter I Representations of the fundamental group and the torsor of deformations. An overview Ahmed Abbes Michel Gros 1

I.1 Introduction 1

I.2 Notation and conventions 3

I.3 Small generalized representations 5

I.4 The torsor of deformations 6

I.5 Faltings ringed topos 13

I.6 Dolbeault modules 19

Chapter II Representations of the fundamental group and the torsor of deformations. Local study Ahmed Abbes Michel Gros 27

II.1 Introduction 27

II.2 Notation and conventions 28

II.3 Results on continuous cohomology of profmitc groups 35

II.4 Objects with group actions 50

II.5 Logarithmic geometry lexicon 63

II.6 Faltings' almost purity theorem 71

II.7 Faltings extension 84

II.8 Galois cohomology 98

II.9 Fontaine p-adic infinitesimal thickenings 110

II.10 Higgs- Tate torsors and algebras- 120

II.11 Galois cohomology II 132

II.12 Dolbeault representations 143

II.13 Small representations 153

II.14 Descent of small representations and applications 166

II.15 Hodge Tate representations 175

Chapter III Representations of the fundamental group of deformations. Global aspects Ahmed Abbes Michel Gros 179

III.1 Introduction 179

III.2 Notation and conventions 180

III.3 Locally irreducible schemes 184

III.4 Adequate logarithmic schemes 185

III.5 Variations on the Koszul complex 190

III.6 Additive categories up to isogeny 194

III.7 Inverse systems of a topos 203

III.8 Partings ringed topos 211

III.9 Faltings topos over a trait 222

III.10 Higgs-Tate algebras 229

111.11 Cohomological computations 250

III.12 Dolbeault modules 266

III.13 Dolbeault modules on a small affine scheme 284

III.14 Inverse image of a Dolbeault module under an étale morpbism 290

III.15 Fibered category of Dolbeault modules 299

Chapter IV Cohomology of Higgs isocrystals Takeshi Tsuji 307

IV.1 Introduction 307

IV.2 Higgs envelopes 313

IV.3 Higgs isocrystals and Higgs crystals 350

IV.4 Cohomology of Higgs isocrystals 369

IV.5 Representations of the fundamental group 383

IV.6 Comparison with Faltings cohomology 402

Chapter V Almost étale coverings Takeshi Tsuji 449

V.1 Introduction 449

V.2 Almost isomorphisms 450

V.3 Almost finitely generated projective modules 452

V.4 Trace 453

V.5 Rank and determinant 455

V.6 Almost flat modules and almost faithfully flat modules 459

V.7 Almost étale coverings 461

V.8 Almost faithfully fiat descent I 464

V.9 Almost faithfully fiat descent II 467

V.10 Liftings 471

V.11 Group cohomology of discrete A-G-modules 478

V.12 Galois cohomology 481

Chapter VI Covanishing topos and generalizations Ahmed Abbes Michel Gros 485

VI.1 Introduction 485

VI.2 Notation and conventions 493

VI.3 Oriented products of topos 494

VI.4 Covanishing topos 502

VI.5 Generalized covanishing topos 511

VI.6 Morphisms with values in a generalized covanishing topos 527

VI.7 Ringed total topos 532

VI.8 Ringed covanishing topos 537

VI.9 Finite étale site and topos of a scheme 542

VI.10 Faltings site and topos 550

VI.11 Inverse limit of Fallings topos 570

Facsimile: A p-adic Simpson correspondence Gerd Faltings, Advances in Mathematics 198 (2005), 847-862 577

Bibliography 595

Indexes 599

Editorial Reviews

"The authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts."—-Gerd Faltings, Zentralblatt MATH

From the Publisher