Stability of unitary \(\text{SK}_1\) of Azumaya algebras.
type: Item
instance of: scholarly article
title: Stability of unitary \(\text{SK}_1\) of Azumaya algebras.
zbMATH Open document ID: 1243.16021
published in: Archiv der Mathematik
publication date: 2012-02-14T00:00:00Z
review text: Let \(A\) be an Azumaya algebra with center \(C\) of rank \(n^2\) and a unitary involution \(\tau\). In the paper under review the author studies the stability of the unitary \(\text{SK}_1\) under reduction. The main result is the following theorem. Theorem 1. Let \(A\) be an Azumaya algebra of rank \(n^2\) over its center \(C\) with a unitary involution \(\tau\) and \(R=C^\tau\), where \(R\) is a Henselian ring and \(2\) and \(n\) invertible in \(R\). Then \(\text{SK}_1(A,\tau)\cong\text{SK}_1(\overline A,\overline\tau)\), where \(\overline A=A/mA\) and \(m\) is the unique maximal ideal of \(R\). Recall that the stability of \(\text{SK}_1\) was previously proved by Platonov for an unramified valued division algebra (over a complete discretely valued field) and by Yanchevskii (over a discretely valued Henselian field). Here, using Theorem 1 above, the author gives another proof of the stability of unitary \(\text{SK}_1\) for the general case of valued division algebras over Henselian fields.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 19B99
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 6006284
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: reduced Whitehead groups
zbMATH Keywords: unitary involutions
zbMATH Keywords: valued division algebras over Henselian fields
author: R. Hazrat
reviewed by: Bùi Xuân Hải
MaRDI profile type: Publication
full work available at URL: s00013-011-0338-y
OpenAlex ID: W1976852110
cites work: Q3963168
cites work: Algebras over nonlocal Hensel rings
cites work: \(\text{SK}_1\) of Azumaya algebras over Hensel pairs.
cites work: \(\text{SK}_1\) of graded division algebras.
cites work: Unitary SK1 of graded and valued division algebras
cites work: Division algebras over Henselian fields.
cites work: Q3997794
cites work: Q3871006
cites work: THE TANNAKA-ARTIN PROBLEM AND REDUCEDK-THEORY
cites work: Q4256372
cites work: Q3153404
DOI: 10.1007/S00013-011-0338-Y
description: scientific article; zbMATH DE number 6006284