p-adic Hodge Theory

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