On Azumaya algebras arising from Clifford algebras
type: Item
instance of: scholarly article
title: On Azumaya algebras arising from Clifford algebras
zbMATH Open document ID: 0652.16002
DOI: 10.1016/0021-8693(88)90224-4
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 4063290
zbMATH Keywords: Clifford algebra
zbMATH Keywords: generalized Clifford algebra
zbMATH Keywords: irreducible factors
zbMATH Keywords: representations
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: affine coordinate ring
zbMATH Keywords: hyperelliptic curve
zbMATH Keywords: Azumaya algebra
zbMATH Keywords: Picard group
zbMATH Keywords: desingularization
zbMATH Keywords: Brauer group
author: Darrell Haile
author: Steven B. Tesser
reviewed by: M.-A. Knus
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90224-4
OpenAlex ID: W2022935782
cites work: On Azumaya algebras and finite dimensional representations of rings
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cites work: Representations of a Clifford algebra that are Azumaya algebras and generate the Brauer group
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description: scientific article; zbMATH DE number 4063290