An isomorphism problem for Azumaya algebras with involution over semilocal Bézout domains.
type: Item
instance of: scholarly article
title: An isomorphism problem for Azumaya algebras with involution over semilocal Bézout domains.
zbMATH Open document ID: 1314.16010
author: Sofie Beke
author: Jan Van Geel
published in: Algebras and Representation Theory
publication date: 2015-01-07T00:00:00Z
review text: Let \(R\) be a semilocal integral domain with quotient field \(K\) and \(2\in R^\times\). One of Grothendieck's conjectures states that for a reductive \(R\)-linear group \(G\), any \(G\)-torsor which is trivial over \(K\) is already trivial over \(R\). In 1989, Nisnevich has proved this for regular local rings \(R\) which are either Henselian or one-dimensional. For a complete discrete valuation ring \(R\), this result was already known to Tits. If \(R\) is a regular local algebra over a field, Panin (2005) proved the conjecture for the automorphism group of an \(R\)-algebra with involution. In the paper under review, the authors prove that for a semilocal Bézout domain \(R\), two \(R\)-algebras with involution are isomorphic whenever the corresponding \(K\)-algebras are isomorphic.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 6384232
zbMATH Keywords: Azumaya algebras with involution
zbMATH Keywords: central simple algebras with involution
zbMATH Keywords: reductive linear algebraic groups
zbMATH Keywords: valuation rings
zbMATH Keywords: semilocal Bézout domains
zbMATH Keywords: skew-Hermitian spaces
zbMATH Keywords: bilinear spaces
zbMATH Keywords: multipliers
MaRDI profile type: Publication
full work available at URL: s10468-013-9463-6
OpenAlex ID: W2041092216
cites work: The Brauer Group of a Commutative Ring
cites work: Overrings of Bezout Domains
cites work: Eine Bemerkung zum schwachen Approximationssatz
cites work: Q5518494
cites work: Q3516073
cites work: Valued Fields
cites work: Q4513600
cites work: Q5595003
cites work: Projective modules over semilocal rings
cites work: Q3997794
cites work: Q4210545
cites work: Q4205555
cites work: Q5703053
cites work: Le théorème d'approximation pour les valuations de Krull
cites work: Azumaya algebras with involution
cites work: Q3277837
cites work: A Cassels-Pfister theorem for involutions on central simple algebras
DOI: 10.1007/S10468-013-9463-6
description: scientific article; zbMATH DE number 6384232