The p-adic Simpson Correspondence (AM-193)

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 f...

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Bibliographic Details
Main Authors:	Abbes, Ahmed, Gros, Michel, Tsuji, Takeshi
Format:	eBook Book
Language:	English
Published:	United States Princeton University Press 2016
Edition:	1
Series:	Annals of Mathematics Studies
Subjects:	Adjoint functors Affine transformation Algebra & number theory Algebraic closure Algebraic Geometry Arithmetical algebraic geometry Automorphism Base change Closed immersion Coefficient Cohomology Cokernel Commutative diagram Commutative property Commutative ring Computation Connected component (graph theory) Corollary Covering space Derived category Diagram (category theory) Direct limit Direct sum Discrete valuation ring Divisibility rule Embedding Endomorphism Equivalence of categories Exact functor Exact sequence Existential quantification Field of fractions Formal scheme Functor Fundamental group Galois cohomology Galois group General Topics for Engineers Generic point Geometry, Algebraic Group theory Higgs bundle Hodge theory Homomorphism Identity element Initial and terminal objects Integer Integral domain Integral element Inverse image functor Inverse limit Inverse system Irreducible component Logarithm Mathematical induction MATHEMATICS MATHEMATICS / Algebra / General MATHEMATICS / General MATHEMATICS / Mathematical Analysis Maximal ideal Monoid Morphism Morphism of schemes p-adic fields p-adic groups P-adic number Presheaf (category theory) Profinite group Rational number Residue field Ring homomorphism Sheaf (mathematics) Spectral sequence Subgroup Summation Tensor product Theorem Topology Torsor (algebraic geometry) Valuation ring Vector bundle Zariski topology Faltings cohomology Koszul complex additive categories finite tale site almost flat module Galois cohomology ringed covanishing topos deformation generalized covanishing topos torsor linear algebra HodgeДate structure small representation fundamental group Dolbeault representation cohomology Faltings ringed topos Hyodo's theory discrete AЇ-module almost faithfully flat module generalized representation almost tale extension crystalline-type topos Higgs bundle HodgeДate representation small generalized representation HodgeДate theory HiggsДate algebra overconvergence locally irreducible scheme Dolbeault module covanishing topos Higgs crystals almost tale covering Faltings extension Faltings topos Higgs bundles Higgs isocrystal Gerd Faltings ringed total topos p-adic Simpson correspondence tale cohomology p-adic field inverse limit solvable Higgs module period ring Faltings site Dolbeault generalized representation tale fundamental group almost faithfully flat descent morphism p-adic Hodge theory Higgs envelopes almost isomorphism adic module P-adic number Rational number Subgroup Spectral sequence Endomorphism Mathematical induction Zariski topology Commutative property Corollary Direct sum Functor Cohomology Topology Integral domain Homomorphism Affine transformation Commutative diagram Existential quantification Torsor (algebraic geometry) Inverse image functor Direct limit Adjoint functors Vector bundle Identity element Integral element Computation Connected component (graph theory) Hodge theory Maximal ideal Summation Exact functor Base change Tensor product Residue field Diagram (category theory) Embedding Ring homomorphism Commutative ring Fundamental group Equivalence of categories Sheaf (mathematics) Cokernel Covering space Logarithm Exact sequence Theorem Formal scheme Valuation ring Galois group Inverse limit Automorphism Field of fractions Coefficient Irreducible component Morphism of schemes Generic point Initial and terminal objects Presheaf (category theory) Profinite group Morphism Inverse system Monoid Discrete valuation ring Derived category Integer Divisibility rule Closed immersion Algebraic closure
ISBN:	1400881234, 9781400881239, 0691170282, 9780691170282, 0691170290, 9780691170299
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Table of Contents:

Front Matter Table of Contents Foreword CHAPTER I: Representations of the fundamental group and the torsor of deformations. CHAPTER II: Representations of the fundamental group and the torsor of deformations. CHAPTER III: Representations of the fundamental group and the torsor of deformations. CHAPTER IV: Cohomology of Higgs isocrystals CHAPTER V: Almost étale coverings CHAPTER VI: Covanishing topos and generalizations Facsimile : Bibliography Indexes
IV.5 Representations of the fundamental group -- IV.6 Comparison with Faltings cohomology -- Chapter V. Almost étale coverings -- V.1 Introduction -- V.2 Almost isomorphisms -- V.3 Almost finitely generated projective modules -- V.4 Trace -- V.5 Rank and determinant -- V.6 Almost flat modules and almost faithfully flat modules -- V.7 Almost étale coverings -- V.8 Almost faithfully flat descent I -- V.9 Almost faithfully flat descent II -- V.10 Liftings -- V.11 Group cohomology of discrete A-G-modules -- V.12 Galois cohomology -- Chapter VI. Covanishing topos and generalizations -- VI.1 Introduction -- VI.2 Notation and conventions -- VI.3 Oriented products of topos -- VI.4 Covanishing topos -- VI.5 Generalized covanishing topos -- VI.6 Morphisms with values in a generalized covanishing topos -- VI.7 Ringed total topos -- VI.8 Ringed covanishing topos -- VI.9 Finite étale site and topos of a scheme -- VI.10 Faltings site and topos -- VI.11 Inverse limit of Faltings topos -- Facsimile : A p-adic Simpson correspondence -- Bibliography -- Indexes
Cover -- Title -- Copyright -- Dedication -- Contents -- Foreword -- Chapter I. Representations of the fundamental group and the torsor of deformations. An overview -- I.1 Introduction -- I.2 Notation and conventions -- I.3 Small generalized representations -- I.4 The torsor of deformations -- I.5 Faltings ringed topos -- I.6 Dolbeault modules -- Chapter II. Representations of the fundamental group and the torsor of deformations. Local study -- II.1 Introduction -- II.2 Notation and conventions -- II.3 Results on continuous cohomology of profinite groups -- II.4 Objects with group actions -- II.5 Logarithmic geometry lexicon -- II.6 Faltings' almost purity theorem -- II.7 Faltings extension -- II.8 Galois cohomology -- II.9 Fontaine p-adic infinitesimal thickenings -- II.10 Higgs-Tate torsors and algebras -- II.11 Galois cohomology II -- II.12 Dolbeault representations -- II.13 Small representations -- II.14 Descent of small representations and applications -- II.15 Hodge-Tate representations -- Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects -- III.1 Introduction -- III.2 Notation and conventions -- III.3 Locally irreducible schemes -- III.4 Adequate logarithmic schemes -- III.5 Variations on the Koszul complex -- III.6 Additive categories up to isogeny -- III.7 Inverse systems of a topos -- III.8 Faltings ringed topos -- III.9 Faltings topos over a trait -- III.10 Higgs-Tate algebras -- III.11 Cohomological computations -- III.12 Dolbeault modules -- III.13 Dolbeault modules on a small affine scheme -- III.14 Inverse image of a Dolbeault module under an étale morphism -- III.15 Fibered category of Dolbeault modules -- Chapter IV. Cohomology of Higgs isocrystals -- IV.1 Introduction -- IV.2 Higgs envelopes -- IV.3 Higgs isocrystals and Higgs crystals -- IV.4 Cohomology of Higgs isocrystals
Contents --
Ahmed Abbes, Michel Gros --
Chapter VI. Covanishing topos and generalizations
Chapter V. Almost étale coverings
Indexes
Foreword --
Chapter IV. Cohomology of Higgs isocrystals
Gerd Faltings --
Chapter II. Representations of the fundamental group and the torsor of deformations. Local study
Facsimile : A p-adic Simpson correspondence
Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects
Frontmatter --
Takeshi Tsuji --
Bibliography --
Chapter I. Representations of the fundamental group and the torsor of deformations. An overview