On Azumaya algebras arising from Clifford algebras

Abstract

In this paper we continue the investigation begun in Haile [4] and Tesser [12] into the structure of the Clifford algebra of a form. If f is a form of degree d in n variables over a field F, then the Clifford algebra off is the F-algebra C,= R/T where R = Fix,, . . . . x,} is the free associative F-algebra in n variables and T is the ideal generated by {(a,x,+ ..’ + %X,Y-f(~1, ---, cx,,)jcxl ,..., cl,~F}. If d=2 this is the classical Clifford algebra of a quadratic form. If d > 2 then this is also called the “generalized” Clifford algebra and has been studied by various authors (see Roby [ 111, Revoy [lo], Childs [2] ). In Haile [4] it was shown that if d= 3 and n = 2 (i.e., iffis a binary cubic form) and the characteristic of the field F is not 2 or 3, then the Clifford algebra C,of the form is Azymaya with center the affine coordinate ring of an explicitly given elliptic curve over F (the center was also determined by Heerema in [6]). Moreover the algebra C, can be extended to an Azumaya algebra over the projective closure of the elliptic curve. Applying the pairing investigated by Lichtenbaum [4] and Manin [9], one obtains then for each field extension K/F a homomorphism from the group of K-rational points on the elliptic curve to the Brauer group of K, given by specialization. In [lo] Revoy, using results of Roby [ 111, exhibited an explicit F-basis for the Clifford algebra of an arbitrary form. Using this basis it is easy to