cohomology

group theory

topos theory

# 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

### 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 ${G}_{\mathrm{Galois}}$ of a field $K$ is the fundamental group of the space $X≔\mathrm{Spec}\left(K\right)$. Hence its delooping $B{G}_{\mathrm{Galois}}$ is the fundamental groupoid

${\Pi }_{1}\left(X\right)\simeq B{G}_{\mathrm{Galois}}\phantom{\rule{thinmathspace}{0ex}}.$\Pi_1(X) \simeq \mathbf{B}G_{Galois} \,.

In cohesive homotopy type theory there exists the fundamental ∞-groupoid-construction

$X:\mathrm{Type}\phantom{\rule{thickmathspace}{0ex}}⊢\phantom{\rule{thickmathspace}{0ex}}\Pi \left(X\right):\mathrm{Type}\phantom{\rule{thinmathspace}{0ex}}.$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 ${G}_{\mathrm{Galois}}$-module $A$ is a $B{G}_{\mathrm{Galois}}$-dependent type;

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

$⊢\phantom{\rule{thickmathspace}{0ex}}\left(\prod _{x:B{G}_{\mathrm{Galois}}}\left(*\to A\right)\right):\mathrm{Type}\phantom{\rule{thinmathspace}{0ex}}.$\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

$x:\Pi \left(X\right)\phantom{\rule{thickmathspace}{0ex}}⊢\phantom{\rule{thickmathspace}{0ex}}A\left(x\right):\mathrm{Type}$x \colon \Pi(X) \;\vdash \; A(x) \colon Type

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

$⊢\phantom{\rule{thickmathspace}{0ex}}\left(\prod _{x:\Pi \left(X\right)}\left(*\to A\right)\right):\mathrm{Type}\phantom{\rule{thinmathspace}{0ex}}.$\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 October 4, 2012 10:55:55 by Urs Schreiber (82.169.65.155)