nLab Introduction to Étale Cohomology

This page collects links related to the textbook

• Günter Tamme,

  Introduction to Étale Cohomology

  1994

on étale cohomology.

Chapter 0 Preliminaries

1. Abelian categories

• abelian category

2. Homological algebra in abelian categories

• homological algebra

2.1 $\delta$-Functors

• delta-functor

2.2 Derived functors

• derived functor

2.3 Spectral sequences

• spectral sequence

• edge morphism

• (2.3.5) Grothendieck spectral sequence

3. Inductive limits

Chapter I Topologies and sheaves

1. Topologies

• Grothendieck topology

2. Abelian presheaves on topologies

2.1 The category of abelian presheaves

• group object

2.2 Cech cohomology

2.3 The functors $f^p$ and $f_p$

• morphism of sites

• direct image, inverse image (here: on presheaves)

3. Abelian sheaves on topologies

3.1 The associated sheaf of a presheaf

• sheafification

3.2 The category of abelian sheaves

• sheaf of abelian groups

3.3 Cohomology of abelian sheaves

• abelian sheaf cohomology

3.4 The spectral sequences for Cech cohomology

• Cech cohomology

3.5 Flabby sheaves

• flabby sheaf

3.6 The functors $f^s$ and $f_s$

• morphism of sites

• direct image, inverse image

3.7 The Leray spectral sequence

• Leray spectral sequence

3.8 Localization

• localization

3.9 The comparison lemma

3.10 Noetherian topologies

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

Chapter II Étale Cohomology

1. The Étale site of a scheme

1.1 Étale morphisms

• étale morphism

1.2 Étale site

• étale site

1.3 Relation between Étale and Zariski cohomology

• relation to Zariski cohomology

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

• direct image, inverse image

• base change

1.5 The restricted Étale site

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

• Galois group

3. Examples of étale sheaves

3.1 Representable sheaves

• sheaf, descent

• étale topos

• representable presheaf

• canonical topology

• additive group, multiplicative group, roots of unity

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

• sheaf of coherent modules

• constructible sheaf

3.3 Big étale site

• big site

4. The theories of Artin-Schreier and of Kummer

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

4.2 The Artin-Schreier sequence

• Frobenius morphism

• Artin-Schreier sequence

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

• Hilbert's theorem 90

• Azumaya algebra, Brauer group

4.4 The Kummer sequence

• Kummer sequence

4.5 The sheaf of divisors on $X_{et}$

5. Stalks of étale sheaves

• stalk

6. Strict localizations

6.1 Henselian rings and strictly local rings

• Henselian ring

• strictly local ring?

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

• torsion sheaf

• constructible sheaf

10. Étale cohomology of curves

11. General theorems in étale cohomology theory

• proper base change theorem

category: reference