nLab
Galois cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Group Theory

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

Galois cohomology is the group cohomology of Galois groups GG. Specifically, for GG the Galois group of a field extension L/KL/K, Galois cohomology refers to the group cohomology of GG with coefficients in a GG-module naturally associated to LL.

Galois cohomology is studied notably in the context of algebraic number theory.

Properties

Relation to étale cohomology

Galois cohomology of a field kk is essentially the étale cohomology of the spectrum Spec(k)Spec(k).

See also at comparison theorem (étale cohomology).

In terms of cohesive homotopy type theory

We make some comments on the formulation of Galois cohomology in cohesive homotopy type theory.

As discussed at Galois theory, the absolute Galois group G GaloisG_{Galois} of a field KK is the fundamental group of the spectrum XSpec(K)X \coloneqq Spec(K). Hence its delooping BG Galois\mathbf{B}G_{Galois} is the fundamental groupoid

Π 1(X)BG Galois. \Pi_1(X) \simeq \mathbf{B}G_{Galois} \,.

In cohesive homotopy type theory there exists the fundamental ∞-groupoid-construction – the shape modality Π\Pi

X:TypeΠ(X):Type. X \colon Type \;\vdash \; \Pi(X) \colon Type \,.

Moreover, by the discussion at group cohomology in the section group cohomology - In terms of homotopy type theory

  1. a G GaloisG_{Galois}-module AA is a BG Galois\mathbf{B}G_{Galois}-dependent type;

  2. the group cohomology is the dependent product over the function type

    ( x:BG Galois(*A)):Type. \vdash \; \left( \prod_{x \colon \mathbf{B}G_{Galois}} \left( * \to A \right)\right) \colon Type \,.

Hence, generally in cohesive homotopy type theory, for XX a type and

x:Π(X)A(x):Type x \colon \Pi(X) \;\vdash \; A(x) \colon Type

a Π(X)\Pi(X)-dependent type, we can say that the corresponding \infty-Galois cohomology is

( x:Π(X)(*A)):Type. \vdash \; \left( \prod_{x \colon \Pi(X)} \left( * \to A\right) \right) \colon Type \,.

Warning. Currently there is not written down yet a model for cohesive homotopy type theory given by a cohesive (∞,1)-topos over a site like the étale site.

Examples

References

  • John Tate, Galois cohomology (pdf)

  • Jean-Pierre Serre, Galois cohomology, Springer Monographs in Mathematics (1997)

    Cohomologie galoisienne, Lecture Notes in Mathematics, 5 (Fifth ed. 1994), Springer-Verlag, MR1324577

  • Grégory Berhuy, An introduction to Galois cohomology and its applications (pdf)

  • M. Kneser, Lectures on Galois cohomology of classical groups (pdf)

  • Wikipedia, Galois cohomology

A generalization in the setup of corings:

Revised on November 22, 2013 01:06:05 by Urs Schreiber (77.251.114.72)