Clifford Algebras and Orthogonal Groups over Commutative Rings

Abstract

The Clifford algebra of a quadratic module M over a form ring (in the orthogonal case with form parameter {0}) is an algebra which is compatible with the structure of M in a universal way. We will study this algebra in this chapter and see that it has important impact on the structure of the orthogonal groups. It will be instrumental not only in clearing up earlier complications that were special to the orthogonal groups but also in generalizing previous results from fields to commutative rings. For example, the Clifford algebra provides a generalized spinor norm which adds insight into the structure of the KO₁ groups.