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