The Brauer Long group of Z/2 dimodule algebras

Abstract

Beginning with the papers of G. Azumaya [2] and M. Auslander and O. Goldman [1] introducing the Brauer group B(R) of a commutative ring R, there have followed a series of extensions of the Brauer group of division algebras over a field to satisfy various needs. A. Grothendeick viewed the Brauer group of a commutative ring as the local part of a Brauer group of schemes and related the Brauer group to the second etale cohomology group of the scheme with values in the units scheaf [11]. At about the same time, C. T. C. Wall introduced in [14] a Brauer group of equivalence classes of Z/2 graded algebras over a field with multiplication induced by a twisted tensor product to study the Witt ring of quadratic forms. Wall’s construction was extended to commutative rings R by H. Bass and C. Small [13], and this group is now called the Brauer Wall group of R and denoted BW(R). A Brauer group of algebras over a field graded by an arbitrary finite abelian group was introduced by M. Knus [9], and extended to arbitrary commutative rings (with a twisted multiplication) by L. Childs, G. Garfinkel and M. Orzech [5]. The Childs-Garfinkel-Orzech construction contained the Brauer Wall group as a special case. An “equivariant Brauer group” of algebras on which a fixed group acted as a group of automorphisms was constructed by O. Frolich and C. T. C. Wall. In his thesis [10], F. W. Long introduced a Brauer group of dimodule algebras on which a grading group G acted as a group of automorphisms which included the affine versions of all the previous extensions of the Brauer group as subgroups. Long’s group is now called the Brauer Long group of R and is denoted BD(R, G). After Long introduced his group, a steady stream of papers have considered the properties and calculations of BD(R, G) and its siblings. Some of these are listed among the references at the end of this report.