The p-adic Simpson correspondence

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