Equivariant group cohomology and Brauer group. (English) Zbl 1036.12002

Cegarra, A. M.; Carzón, A. R.

Bull. Belg. Math. Soc. - Simon Stevin 10, No. 3, 451-459 (2003).

Let \(\Gamma\) be a fixed group of operators. By a \(\Gamma\)-group \(G\) we mean a group \(G\), endowed with a left \(\Gamma \)-action by automorphisms, \((\sigma , x) \to ^{\sigma} x\), for all \(\sigma \in \Gamma\), \(x \in G\). A \(\Gamma\)-equivariant module over a \(\Gamma \)-group \(G\) means a \(\Gamma\)-module \(A\), i.e. an abelian \(\Gamma\)-group, provided with a structure of a \(G\)-module, such that \(^ {\sigma} (^ x a) = ^{^ {\sigma} x} (^ x a)\): \(\sigma \in \Gamma\), \(x \in G\) and \(a \in A\). When this occurs, we say that the \(\Gamma\)-action map \(G \times A \to A\) is \(\Gamma\)-equivariant. Suppose now that \(F/K\) is a finite Galois field extension, \(G\) is its Galois group, and an action of \(\Gamma\) on \(F\) by \(K\)-automorphisms is given. The paper under review shows that then there is an isomorphism, \(H _ r ^ 2 (G, F ^ {\ast }) \cong \) Br\(_ r (F/K)\), between the second equivariant cohomology group of the \(\Gamma\)-group \(G\) (where the \(\Gamma\)-action is by conjugation) with coefficients in the \(\Gamma\)-equivariant \(G\)-module \(F ^ {\ast }\) (with the natural actions of \(\Gamma\) and \(G\)) and the group of equivariant isomorphism classes of finite dimensional central simple \(K\)-algebras endowed with a \(\Gamma\)-action by \(K\)-automorphisms and containing \(F\) as a \(\Gamma\)-equivariant strictly maximal subfield. Its main result is presented in a more generalized form.

Reviewer: Ivan D. Chipchakov (Sofia)

MSC:

12G05	Galois cohomology
16H05	Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16K50	Brauer groups (algebraic aspects)
20J06	Cohomology of groups

Keywords:

group cohomology; Brauer group; Azumaya algebra; Galois extension; group of operators