Consequences of the Existence of Special Elements

Overview

The purpose of this chapter is the investigation of the consequences that the existence of special elements has for the structure of C(M), particularly as to the interplay between C(M) and its subalgebra C₀(M). If rank M is odd, we will see that the structure of C₀(M) completely determines that of C(M); if rank M is even, this situation is reversed, and it is C(M) that determines C₀(M). Throughout the chapter, R is a commutative ring and M is a quadratic module over R with quadratic form q and associated symmetric bilinear form h. An isomorphism between algebras for which gradings are specified will be understood to preserve the gradings. For an R-algebra A, recall that <A> denotes the algebra A supplied with the trivial grading.