Azumaya algebras with involutions
type: Item
instance of: scholarly article
title: Azumaya algebras with involutions
zbMATH Open document ID: 0695.16003
DOI: 10.1016/0021-8693(90)90100-3
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let R be a commutative ring and S/R an étale quadratic extension. Br(-) is the Brauer group and \({}_ 2Br(\)-) its 2-torsion. This paper studies the problem of when the sequence \[ _ 2Br(R)\quad \to^{Res}\quad_ 2Br(S)\quad \to^{Cores}\quad_ 2Br(R) \] is exact. If R, S are fields and char \(R\neq 2\), the sequence is known to be exact. From the authors' introduction: ``Let A be an Azumaya S-algebra whose class [A] belongs to kernel (Cores: \({}_ 2Br(S)\to_ 2Br(R))\). For [A] to belong to \(Res(_ 2Br(R))\), it is necessary and sufficient that there exists an Azumaya S-algebra B Brauer equivalent to A admitting involutions of first and second kinds which commute. Results of Saltman assert that there exist algebras \(B_ i\), \(i=1,2\), Brauer equivalent to A which support involutions of the first and second kind, respectively. We construct a homomorphism \[ h: \ker (Cores:_ 2Br(S)\to_ 2Br(R))\to Pic(R)/Cores Pic(S) \] whose kernel consists precisely of classes of Azumaya S-algebras which support commuting involutions of both kinds. Thus, h is a measure of the obstruction to the sequence being exact... Finally, we give an example of an étale quadratic extension S/R for which (the sequence) is not exact.''
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 4137973
zbMATH Keywords: étale quadratic extension
zbMATH Keywords: Brauer group
zbMATH Keywords: Azumaya S-algebra
zbMATH Keywords: involutions
zbMATH Keywords: Brauer equivalent
author: M.-A. Knus
author: R. Parimala
author: V. Srinivas
reviewed by: Margaret Beattie
MaRDI profile type: Publication
full work available at URL: 0021-8693(90)90100-3
OpenAlex ID: W2043073725
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description: scientific article; zbMATH DE number 4137973