Structure of Clifford and Arf Algebras

Overview

Let M be a finitely generated projective nonsinguiar quadratic module over a commutative ring. This chapter establishes the basic structure theory of the Clifford algebra C(M) and its subalgebras C₀(M), A(M), Cen C(M) and Cen C₀(M). It will be proved that both C(M) and C₀(M) are separable. For a faithful M it will be shown that the Arf algebra A(M) is separable quadratic, and that the following equivalences hold:

$$ CenC(M) = R \Leftrightarrow rank\;M\;is\;even \Leftrightarrow Cen\,{C_{0}}(M) = A(M) $$

and

$$ Cen{C_{0}}(M) = R \Leftrightarrow rank\;M\;is\;odd \Leftrightarrow Cen\,C(M) = A(M) $$

.