The equivariant Brauer group of a group

Abstract

We consider the Brauer group BM'(k, G) of a group G (finite or infinite) over a commutative ring k with identity. A split exact sequence 1 → Br'(k) → BM'(k, G) → Gal(k, G) → 1 is obtained. This generalizes the Frohlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of G to the infinite case of G. Here Br'(k) is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.