The Brauer-Wall group of a commutative ring

Abstract:

Let k be a commutative ring (with 1). We work with k-algebras with a grading $\bmod \;2$, and with graded modules over such algebras. Using graded notions of tensor product, commutativity, and morphisms, we construct an abelian group ${\rm {BW}}(k)$ whose elements are suitable equivalence classes of Azumaya k-algebras. The consruction generalizes, and is patterned on, the definition of the Brauer group ${\rm {Br}}(k)$ given by Auslander and Goldman. ${\rm {Br}}(k)$ is in fact a subgroup of ${\rm {BW}}(k)$, and we describe the quotient as a group of graded quadratic extensions of k.