On Azumaya algebras arising from Clifford algebras
The authors continue the investigation begun by \textit{D. Haile} [Am. J. Math. 106, 1269-1280 (1984; Zbl 0585.13001)] and \textit{S. Tesser} [Representations of a Clifford algebra that are Azumaya algebras and generate the Brauer group, J. Algebra 119, No.2, 265-281 (1988)]. Let \(C_ f\) be the generalized Clifford algebra of a form of degree d in n variables. In the first section they show that if f has no repeated irreducible factors over the algebraic closure of the ground field, then every representation of \(C_ f\) of degree d is Azumaya. In section two it is shown that for many of the representations found in the paper of Tesser the image of the Clifford algebra can be given by an explicit finite set of generators and relations. In section three the case of a diagonal binary form is considered in detail. By Tesser there exists a representation which is Azumaya and with center the affine coordinate ring of an explicitly given hyperelliptic curve. These curves have singularities at infinity. It is shown that the Azumaya algebra can be extended to the desingularization of the projective closure of the curve. As a consequence the authors construct a homomorphism from the Picard group of the desingularization to the Brauer group of the ground field.
scientific article; zbMATH DE number 4063290
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