Computing the Brauer group of graded Azumaya algebras from its subgroups

Abstract

Let R be a connected commutative ring with unity and G a finite abelian group of order n and exponent m. Assume that n is a unit in R, Pit,(R) = 0 and R contains a primitive m th root of unity. In [9], Long defined BD(R, G), a generalized Brauer group of Gdimodule algebras, i.e., algebras with a G-grading upon which G acts as a group of grade-preserving automorphisms. This is a generalization of the Brauer-Wall group [ 17, 151, and the graded Brauer groups of Knus [S] and Childs, Garlinkel, and Orzech [4]. Just as Clifford algebras are elements of the Brauer-Wall group, so the generalized Clifford algebras of [ 12, 13, 161 are elements of Long’s generalized Brauer group. BD(R, G) was first computed for R a separably closed field and G cyclic of order a product of primes; later BD(R, G) was computed for any cyclic G and more general R, [9, 14, 2, 31. The key to these computations was the normal subgroup of central G-Azumaya algebras [2]. However, as Orzech pointed out in [ 141, if G is noncyclic, the set of central G-Azumaya algebras may not be a subgroup. Therefore, to compute BD(R, G) for noncyclic G, one must try a new approach. One approach is that taken by Childs [S] in which he studies a group of graded Galois extensions, Galz,(R, G), defined to provide an image of B&R, G) under a map rt with kernel the usual Brauer group of R. If G z G*, then BD(R, G) % B,(R, G x G) for a particular bilinear map 4. He finds that for G a direct product of cyclic groups of order pp, BD(R, G) is described by the exact sequences