p-adic Hodge Theory

Description

Preface Contents Notes on the mathbbAinf-Cohomology of Integral p-Adic Hodge Theory 1 Introduction 1.1 Mysterious Functor and Crystalline Comparison 1.2 Statement of Main Theorem and Outline of Notes 2 The décalage Functor Lη: Modifying Torsion 2.1 Example 1: Crystalline Cohomology 2.2 ``Example 2'': An Integral Form of Faltings' Almost Purity Theorem 3 Algebraic Preliminaries on Perfectoid Rings 3.1 The Maps θr, tildeθr 3.2 Perfectoid Rings 3.3 Main Example: Perfectoid Rings Containing Enough Roots of Unity 4 The Pro-étale Site and Its Sheaves 4.1 The Pro-étale Site Xprot 4.2 More Sheaves on Xprot 4.3 Calculating Pro-étale Cohomology 5 The Main Construction and Theorems 6 Witt Complexes 6.1 Langer–Zink's Relative de Rham–Witt Complex 6.2 Constructing Witt Complexes 6.3 The de Rham–Witt Complex of a Torus as Group Cohomology 7 The Proof of the p-Adic Cartier Isomorphism 7.1 Technical Lemmas: Base Change and Global-to-Local Isomorphisms 7.2 Reduction to a Torus and to Theorem 10 References On the Cohomology of the Affine Space 1 Introduction 2 Syntomic Variations 3 Computation of HKir(An) 4 Computation of DRri(An) 5 Proof of Theorems1 and 3 5.1 Algebraic Isomorphism 5.2 Topological Considerations References Arithmetic Chern–Simons Theory II 1 The Arithmetic Chern–Simons Action: Introduction and Definition 2 The Arithmetic Chern–Simons Action: Boundaries 3 The Arithmetic Chern–Simons Action: The p-adic Case 4 Towards Computation: The Decomposition Formula 5 Examples 5.1 General Strategy 5.2 Trivialisation of a Pullback of ε 5.3 Local Invariant Computation 5.4 Construction of Examples 5.5 Case 1: Cyclic Group 5.6 Case 2: Non-cyclic Abelian Group 5.7 Case 3: Non-abelian Group 6 Application 7 Appendix 1: Conjugation on Group Cochains 8 Appendix 2: Conjugation Action on Group Cochains: Categorical Approach 8.1 Notation 8.2 Idea 8.3 Cohomology of Categories 8.4 Definition of the Cochains ha,f 8.5 Composing Natural Transformations 8.6 Explicit Formula for ha1,…,ak,f References Some Ring-Theoretic Properties of A`3́9`42`"̇613A``45`47`"603Ainf 1 Finite Generation Properties 2 Vector Bundles 3 Adic Glueing References Sur une q-déformation locale de la théorie de Hodge non-abélienne en caractéristique positive 1 Introduction 2 Rappels sur la théorie d'Ogus et Vologodsky 3 Opérateurs différentiels q-déformés 4 p-courbure et Frobenius divisé q-déformés 5 Théorie de Hodge non-abélienne q-déformée 6 Questions-Travaux en cours References Crystalline mathbbZp-Representations and Ainf-Representations with Frobenius 1 Introduction 2 Period Rings 3 Filtered Crystals 4 The Relative Fontaine–Laffaille Theory by Faltings 5 Acrys-Representations with and Fil 6 Filtered -Modules 7 Filtered (,G)-Modules 8 Ainf-Representations with 9 Duality for Ainf/πp-1-Representations with 10 Period Map 11 Fully Faithfulness of Tcrys and Ainf-Representations with 12 Period Rings Associated to a Framing 13 Acrys-Representations with and Fil 14 Preliminaries on Décalage Functor and Continuous Group Cohomology 15 Galois Cohomology of Ainf-Representations and de Rham Complexes 16 Comparison Theorem with de Rham Complex over Ainf/π 17 Period Rings with Truncated Divided Powers 18 Period Rings with Truncated Divided Powers Associated to a Framing 19 de Rham Complexes with Truncated Divided Powers 20 Comparison Morphism from de Rham Complex over Acrys 21 Comparison Theorem with de Rham Complex over Acrys References