Algebras with Standard Involution

Overview

Again, R is any commutative ring. The goal of this chapter is the analysis of the structure of an algebra over R with “standard” involution. The Clifford algebra C(M) of a quadratic module M over R which is nonsingular and free of rank 2 is the most prominent example and will receive particular attention. A number of the concepts and constructions of the previous chapter are illustrated in the process. In addition, we will see that C(M) is separable over R, that the center of C(M) is R, and that Cen C₀(M) = C₀(M) is a free separable quadratic algebra over R. These matters will be taken up for a general non-singular M in Chapter 9. The special case of rank 2 is a cornerstone for this investigation.