nLab Galois cohomology





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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.


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.



See also:

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.

Last revised on September 2, 2021 at 08:58:09. See the history of this page for a list of all contributions to it.