Introduction to Étale Cohomology




Special and general types

Special notions


Extra structure



Étale morphisms

This page collects links related to the textbook

on étale cohomology.


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.