Representations of a Clifford algebra that are Azumaya algebras and generate the Brauer group

Abstract

Clifford algebras of forms of degree d > 2 are infinite dimensional (see Theorem 3, p. 272 of L. Childs, J. Lin. Mul. Alg., 5 (1978), 267–278 and Corollary 2, p. 273 of M. Orzech and L. Small, Lecture Notes in Pure and Applied Mathematics, Vol. 11, Dekker, New York, 1975). In the case of a non-singular binary cubic form over field F, D. Haile ( Amer. Math. J., 106, No. 6 (1984) , 1269–1280) demonstrated an injective representation of the Clifford algebra in M(3, K), where K is a transcendental extension of F; he showed that the Clifford algebra is thus an Azumaya algebra over the coordinate ring of an elliptic curve, and that, for any algebraic extension F ∼ of F, the group of F ∼ -points on the curve map homomorphically into the Brauer group of F ∼.