The Brauer Group of Dimodule Algebras

In this paper we develope a Brauer group theory for algebras which are simultaneously H-modules and H-comodules, where H is a Hopf algebra. This work generalizes, and also arises naturally from, the work in [4] where we considered algebras acted on by a group and graded by the same group.
In Section 1 we consider the case where there is only an H-module action. This generalizes the equivariant Brauer group of Frohlich and Wall [2] and at the end of Section 1 we show that some of Frijhlich and Wall’s results extend to the Hopf algebra situation.
H-comodule action is studied in Section 2, and we show that this is a straightforward generalization of group grading. The two concepts of module and comodule action are applied simultaneously in Section 3. When they commute (in a sense made precise in Section 3) we call the resulting structure an H-dimodule. This is not the same as an H Hopf module (see [6]) as the commuting condition is different.
In Section 4 we say when an H-dimodule algebra is H-Azumaya. This generalizes the usual definition of Azumaya algebra to our situation and allows us to define “the Brauer group of dimodule algebras.” We then obtain some of its properties.
We conclude the paper with an example. In Section 5 we consider the Brauer group of dimodule algebras over a field of characteristic p where the Hopf algebra is the group ring over that field of the group of p elements. This covers the outstanding case left from [4]. Throughout, R is a fixed commutative ring with 1, each 0, Horn, etc. is taken over R and each map is R-linear unless otherwise stated.