nLab
Introduction to Étale Cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Étale morphisms

This page collects links related to the textbook

on étale cohomology.

Contents

Chapter 0 Preliminaries

1. Abelian categories

2. Homological algebra in abelian categories

2.1 δ\delta-Functors

2.2 Derived functors

2.3 Spectral sequences

3. Inductive limits

Chapter I Topologies and sheaves

1. Topologies

2. Abelian presheaves on topologies

2.1 The category of abelian presheaves

2.2 Cech cohomology

2.3 The functors f pf^p and f pf_p

3. Abelian sheaves on topologies

3.1 The associated sheaf of a presheaf

3.2 The category of abelian sheaves

3.3 Cohomology of abelian sheaves

3.4 The spectral sequences for Cech cohomology

3.5 Flabby sheaves

3.6 The functors f sf^s and f sf_s

3.7 The Leray spectral sequence

3.8 Localization

3.9 The comparison lemma

3.10 Noetherian topologies

3.11 Commutation of the functors H q(U,)H^q(U,-) with pseudofiltered inductive limits

Chapter II Étale Cohomology

1. The Étale site of a scheme

1.1 Étale morphisms

1.2 Étale site

1.3 Relation between Étale and Zariski cohomology

1.4 The functors f *f_\ast and f *f^\ast

1.5 The restricted Étale site

2. The case X=Spec(k)X = Spec(k)

3. Examples of étale sheaves

3.1 Representable sheaves

3.2 Étale sheaves of 𝒪 X\mathcal{O}_X-modules

3.3 Big étale site

4. The theories of Artin-Schreier and of Kummer

4.1 The groups H q(X,(𝔾 a) X)H^q(X, (\mathbb{G}_a)_X)

4.2 The Artin-Schreier sequence

4.3 The groups H q(X,(𝔾 m) X)H^q(X, (\mathbb{G}_m)_X)

4.4 The Kummer sequence

4.5 The sheaf of divisors on X etX_{et}

5. Stalks of étale sheaves

6. Strict localizations

6.1 Henselian rings and strictly local rings

6.2 Strict localization of a scheme

6.3 Étale cohomology on projective limits of schemes

7. The Artin spectral sequence

  • Artin spectral sequence?

8. The decomposition theorem. Relative cohomology

9. Torsion sheaves, locally constant sheaves, constructible sheaves

10. Étale cohomology of curves

11. General theorems in étale cohomology theory

category: reference

Last revised on November 24, 2013 at 12:16:08. See the history of this page for a list of all contributions to it.