p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects

Preface
Contents
List of Contributors
The Relative Hodge–Tate Spectral Sequence: An Overview
	1 Introduction
	2 The Local Version of the Relative Hodge–Tate Spectral Sequence
	3 The Global Version of the Relative Hodge–Tate Spectral Sequence
	4 Faltings\' Main p-adic Comparison Theorems
	5 Relative Faltings Topos
	References
Semisimplicity of the Frobenius Action on π1
	1 Introduction
		1.1 Canonical Splittings
			Applications
	2 Weil–Deligne Representations
		2.1 The Canonical Splitting
		2.2 Mixed Representations as Deformations of Pure Representations
		2.3 The Tannaka Group of Mixed Weil–Deligne Representations
	3 Results on Semisimplicity and Weight–Monodromy
		3.1 The étale Fundamental Group
		3.2 The Crystalline Setting
		3.3 The Main Theorems
			Preliminaries
		3.4 Weight-Monodromy for π1
		3.5 Semisimplicity
	4 Structure of Local Bloch–Kato Selmer Schemes
		4.1 Torsors in Tannakian Categories
			Torsors in Mixed Weil–Deligne Representations
	Appendix: A Canonical Presentation for the Weight-Graded Fundamental Group
		The Complex Case
		The General Case
	References
Logarithmic Adic Spaces: Some Foundational Results
	1 Introduction
		1.1 Notation and Conventions
	2 Log Adic Spaces
		2.1 Recollection on Monoids
		2.2 Log Adic Spaces
		2.3 Charts and Fiber Products
	3 Log Smoothness and Log Differentials
		3.1 Log Smooth Morphisms
		3.2 Log Differentials
		3.3 Sheaves of Log Differentials
	4 Kummer étale Topology
		4.1 The Kummer étale Site
		4.2 Abhyankar\'s Lemma
		4.3 Coherent Sheaves
		4.4 Descent of Kummer étale Covers
		4.5 Localization and Base Change Functors
		4.6 Purity of Torsion Local Systems
	5 Pro-Kummer étale Topology
		5.1 The Pro-Kummer étale Site
		5.2 Localization and Base Change Functors
		5.3 Log Affinoid Perfectoid Objects
		5.4 Structure Sheaves
	6 Kummer étale Cohomology
		6.1 Toric Charts Revisited
		6.2 Primitive Comparison Theorem
		6.3 p-Adic Local Systems
		6.4 Quasi-Unipotent Nearby Cycles
	A Kiehl\'s Property for Coherent Sheaves
	References
Twisted Differential Operators and q-Crystals
	Introduction
	1 δ-Structures
	2 δ-Rings and Twisted Divided Powers
	3 q-Divided Powers and Twisted Divided Powers
	4 Complete q-PD-Envelopes
	5 Hyper q-Stratifications
	6 q-Calculus
	7 q-Crystals
	References
Full Level Structures on Elliptic Curves
	1 Overview
	2 Level Structures on Elliptic Curves
	3 The Ordinary Locus
	4 Kunz\'s Theorem in Mixed Characteristic
	References
The Saturated de Rham–Witt Complex for Schemes with Toroidal Singularities
	Introduction
	1 Dieudonné Complexes and Dieudonné Algebras
	2 Dieudonné Complexes of Monoid Algebras
	3 Idealized Monoid Algebras
	4 Ideally Toroidal Schemes
	5 Crystalline Cohomology
	6 Log Schemes
	7 The Hodge and Nygaard Filtrations
	A Technicalities of Toric Differentials
	References