Arf Algebras and Special Elements

Overview

Let R be any commutative ring. Let M be a quadratic module over R and let C(M) be its Clifford algebra. The centralizer of the even subalgebra C₀(M) in C(M) is a graded algebra which carries important information. We will see in Chapter 8 that it controls the relationship between the structures of C(M) and C₀(M), provides the connection between the tensor product and graded tensor product of Clifford algebras, and that it has consequences for the representations of C(M). The (graded) isomorphism class of this centralizer leads to an invariant for quadratic forms over R, which has significant impact on their structure. This is pursued in Chapters 13 and 14. Since this invariant reduces to the classical Arf invariant when R is a field of characteristic 2, we will denote the centralizer by A(M) and call it the Arf algebra of M. (It is also known as the discriminant algebra of M in the literature.) The present chapter develops some of the very basic properties of A(M). It is assumed throughout that M is finitely generated projective and nonsingular. The quadratic and bilinear forms of M are denoted by q and h, respectively.