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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Hlavní autoři:	Abbes, Ahmed, Gros, Michel, Tsuji, Takeshi
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	United States Princeton University Press 2016
Vydání:	1
Edice:	Annals of Mathematics Studies
Témata:	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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Popis
Shrnutí:	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.
Bibliografie:	Includes bibliographical references and indexes
ISBN:	1400881234 9781400881239 0691170282 9780691170282 0691170290 9780691170299
DOI:	10.1515/9781400881239