ABSTRACT
In this section we use the notation introduced in 1.1.2. In [96], F. Long introduced a Brauer group of a commutative ring with respect to a finite abelian group G. This group generalized, in a sense, the Brauer-Wall group [101, 107], the graded Brauer group of Childs, Garfinkel, and Orzech [86, 87], and the equi-variant Brauer group of Fröhlich and Wall [91]. We shall apply the techniques of graded cohomology, as developed in the Z-graded case in Chapters III, IV, and V to this situation; as the reader may expect, the cohomology groups of the group G will show up in the picture. We shall obtain a cohomological description of Long’s Brauer group, or at least of some of its subgroups; this will yield some cohomological proofs of classical results concerning the Brauer-Long group (cf. [77–81, 89, 90, 96, 98, 99]).
