nLab Galois cohomology

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 $Spec(k)$.

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 $G_{Galois}$ 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.

Examples

• Hilbert's theorem 90

Related concepts

• algebraic number theory

Literature

• John Tate, Galois cohomology (pdf)

• Jean-Pierre Serre, Galois cohomology, Springer Monographs in Mathematics (1997) doi:10.1007/978-3-642-59141-9

  Cohomologie galoisienne, Lecture Notes in Mathematics, 5 (Fifth ed. 1994), Springer-Verlag, MR1324577 doi:10.1007/BFb0108758)

• Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press 2006 (doi:10.1017/CBO9780511607219, pdf)

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

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

See also:

• Wikipedia, Galois cohomology

A generalization in the setup of corings:

• Tomasz Brzezi?ski?, Descent cohomology and corings, Comm Algebra 36:1894-1900, 2008, math.RA/0601491

In a model theoretic context:

• Anand Pillay, Remarks on Galois cohomology and definability, The Journal of Symbolic Logic 62:2 (1997) 487-492 doi
• B. Poizat, Une theorie de Galois imaginaire, J. Symb. Logic 48 (1983) 1151-1170.