The Brauer-Wall group of a commutative ring

Let k be a commutative ring (with 1). We work with ¿-algebras with a grading mod 2, and with graded modules over such algebras. Using graded notions of tensor product, commutativity, and morphisms, we construct an abelian group BW (k) whose elements are suitable equivalence classes of Azumaya ¿-algebras. The consruction generalizes, and is patterned on, the definition of the Brauer group Br (k) given by Ausländer and Goldman. Br (k) is in fact a subgroup of BW (k), and we describe the quotient as a group of graded quadratic extensions of k. Introduction. The subject of this paper is a functor, BW, from commutative rings to abelian groups. The Brauer group Br ik) introduced by Auslander and Goldman in [1] is a subgroup of the Brauer-Wall group BW ik), and the construction of BW is patterned on the construction of Br. The "enlarged" Brauer group was considered for fields by C. T. C. Wali in [9], and the structure theory of [9], presented in §5 below, is an important tool in our work here. Also among the principal sources for this paper we call attention to the Tata notes [3] of H. Bass. Chapter III of [3] presents much of the basic material of [1]; Chapter II presents the "Morita theory" which, suitably translated into our graded context, is another important tool for us ; and the construction of BW is outlined in Chapter IV. The original motivation for enlarging the Brauer group, in [3] as in [9], came from the study of quadratic forms. Various inconveniences in this theory are eliminated if the Clifford algebra of a form is viewed as an element of BW ik) rather than Br ik). The Brauer-Wall group is therefore a fundamental tool towards a general theory of quadratic forms (over an arbitrary commutative ring). This theme is developed in [3, Chapter V]. A second motivation comes from algebraic Zsf-theory. Karoubi has shown (in [6] and a series of subsequent papers) that Clifford algebras play a dramatic unifying role among the various ZC-theories of topology. One hopes that they, and Azumaya algebras generally, may similarly provide the key to a more Received by the editors July 7, 1970. AMS 1970 subject classifications. Primary 13A20, 16A16; Secondary 13B05, 18F25.