The p-adic Simpson Correspondence (AM-193) (Annals of Mathematics Studies, 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 following two new approaches. It mainly focuses 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 book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book 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. Contents Foreword 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 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 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 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 IV.5 Representations of the fundamental group IV.6 Comparison with Faltings cohomology 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.11 Group cohomology of discrete A-G-modules V.10 Liftings V.12 Galois cohomology 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 1. Introduction 2. Generalised representations 3. The local structure of generalised representations 4. Globalisation 5. Examples and open questions References Bibliography Indexes A P-adic Simpson Correspondence II: Small Representations 1. Introduction 2. Berger's rings in higher dimensions 3. Bundles with a singular connection 4. Examples References "The irresistible enthusiasm of Great Adaptations couldn't come at a better time."—David P. Barash, Wall Street Journal "Be very amazed."—Carl Safina, author of Beyond Words and Becoming Wild How one scientist unlocked the secrets behind some of nature's most astounding animals From star-nosed moles that have super-sensing snouts to electric eels that paralyze their prey, animals possess unique and extraordinary abilities. In Great Adaptations , Kenneth Catania presents an entertaining and engaging look at some of nature's most remarkable creatures. Telling the story of his biological detective work, Catania sheds light on the mysteries behind the behaviors of tentacled snakes, tiny shrews, zombie-making wasps, and more. 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Examining some strange and spectacular creatures, Great Adaptations offers a wondrous journey into nature's grand designs. 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{u2014}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 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. Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical work of the twentieth century. The series continues this tradition as Princeton University press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are again available in paperback. For a complete list of titles, please visit the Princeton University Press website: press.princeton.edu. The most recently published volumes include