nLab Galois cohomology

group theory

Cohomology and Extensions

Topos Theory

Could not include topos theory - contents

Contents

Idea

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

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

Properties

Relation to étale cohomology

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

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 $G_{Galois}$ of a field $K$ is the fundamental group of the spectrum $X \coloneqq Spec(K)$. Hence its delooping $\mathbf{B}G_{Galois}$ is the fundamental groupoid

$\Pi_1(X) \simeq \mathbf{B}G_{Galois} \,.$

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

$X \colon Type \;\vdash \; \Pi(X) \colon Type \,.$

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

1. a $G_{Galois}$-module $A$ is a $\mathbf{B}G_{Galois}$-dependent type;

2. the group cohomology is the dependent product over the function 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 $X$ a type and

$x \colon \Pi(X) \;\vdash \; A(x) \colon Type$

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

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

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.

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)