Special and general types
Cohomology and Extensions
Cohomology and homotopy
In higher category theory
Galois cohomology is the group cohomology of Galois groups . Specifically, for the Galois group of a field extension , Galois cohomology refers to the group cohomology of with coefficients in a -module naturally associated to .
Galois cohomology is studied notably in the context of algebraic number theory.
Relation to étale cohomology
Galois cohomology of a field is essentially the étale cohomology of the spectrum .
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 of a field is the fundamental group of the spectrum . Hence its delooping is the fundamental groupoid
In cohesive homotopy type theory there exists the fundamental ∞-groupoid-construction – the shape modality –
Moreover, by the discussion at group cohomology in the section group cohomology - In terms of homotopy type theory
a -module is a -dependent type;
the group cohomology is the dependent product over the function type
Hence, generally in cohesive homotopy type theory, for a type and
a -dependent type, we can say that the corresponding -Galois cohomology is
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.
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