The equivariant Brauer group of a group

Caenepeel, S.; Van Oystaeyen, F.; Zhang, Y.

doi:10.1090/S0002-9939-05-08041-X

The equivariant Brauer group of a group
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by S. Caenepeel, F. Van Oystaeyen and Y. H. Zhang PDF

Proc. Amer. Math. Soc. 134 (2006), 959-972 Request permission

Abstract:

We consider the Brauer group ${\operatorname {BM}’}(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence \[ 1\longrightarrow \operatorname {Br}’(k)\longrightarrow \operatorname {BM}’(k,G)\longrightarrow \operatorname {Gal}(k,G) \longrightarrow 1\] 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 $\operatorname {Br}’(k)$ 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.

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