Reduced norms in the K-theory of orders
type: Item
instance of: scholarly article
title: Reduced norms in the K-theory of orders
zbMATH Open document ID: 0358.16021
DOI: 10.1016/0021-8693(77)90390-8
published in: Journal of Algebra
publication date: 1977-01-01T00:00:00Z
review text: Let \(A\) be an \(R\)-order in a semisimple algebra \(B\). In this paper general formulae of an idelic nature are obtained for the subgroups \(Cl(A)\) and \(G^{f_0^l}(A)\) of \(K_0(A)\) and \(G_0(A)\) consisting of those elements killed by every map got from localization at the maximal ideals of \(R\). These formulae are similar to, and include, that published by several authors for \(Cl(A)\) in the arithmetic case. The kernel \(D(A)\) of the extension map from \(Cl(A)\) to \(Cl(M)\), where \(M\) is a maximal order containing \(A\), is also considered. It is rapidly deduced that if \(B\) is simple (that \(A\) contains all central idempotents is enough) then \(D(A)\) is the kernel of the Cartan map from \(Cl(A)\) to \(G^{f_0^l}(A)\) and that if \(A\) is hereditary then \(D(A)\) is \(0\). The cokernel of the Cartan map is also considered. The author concludes with a demonstration of the idelic approach to class groups. After proving some results about reduced norms he shows that the 2-part of the class group of the quaternion group of order \(4^{\ell r}\) (\(\ell\) prime) is elementary abelian of order \(2^r\) (he relies on a result of Fröhlich that the class group has a quotient of this type).
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 3557996
author: Stephen M. J. Wilson
reviewed by: Stephen M. J. Wilson
MaRDI profile type: Publication
full work available at URL: 0021-8693(77)90390-8
OpenAlex ID: W2091316211
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description: scientific article; zbMATH DE number 3557996