The equivariant Brauer group of a group

Abstract

We consider the Brauer group ${\mathrm{BM}}^{\prime }(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence $$1⟶{\mathrm{Br}}^{\prime }(k)⟶{\mathrm{BM}}^{\prime }(k,G)⟶\mathrm{Gal}(k,G)⟶1$$ is obtained. This generalizes the Fröhlich-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 ${\mathrm{Br}}^{\prime }(k)$ 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.

13 pages