K-theory of group-rings of finite groups over maximal orders in division algebras
type: Item
instance of: scholarly article
title: K-theory of group-rings of finite groups over maximal orders in division algebras
zbMATH Open document ID: 0548.16009
DOI: 10.1016/0021-8693(84)90122-4
author: Aderemi O. Kuku
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let \(\pi\) be a finite group, R the ring of integers in a p-adic field F and A a maximal R-order in a central division algebra D over F. By a result of \textit{R. G. Swan} [Topology 2, 85-110 (1963; Zbl 0119.029)] there is an isomorphism \(G_ 0(R\pi)\to G_ 0(F\pi).\) The author shows that this result does not extend to the non-commutative setting, i.e. the map \(G_ 0(A\pi)\to G_ 0(D\pi)\) is not always injective. This observation then leads to various finiteness theorems about the kernel of \(G_ n(A\pi)\to G_ n(D\pi)\) for \(n\geq 0\) both in the local and in the global case. We only mention the following more general result, which settles a question posed by Swan: Let R be the ring of integers in a number field F and \(\Lambda\) any R-order in a separable F-algebra \(\Sigma\). For all \(n\geq 0\) the group \(G_ n(\Lambda)\) is finitely generated and the kernel of \(G_ n(\Lambda)\to G_ n(\Sigma)\) is a finite group, which is trivial for even n.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Kxx
zbMATH DE Number: 3873528
zbMATH Keywords: finite group
zbMATH Keywords: ring of integers
zbMATH Keywords: p-adic field
zbMATH Keywords: maximal R-order
zbMATH Keywords: central division algebra
zbMATH Keywords: finiteness theorems
zbMATH Keywords: number field
zbMATH Keywords: separable F-algebra
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90122-4
OpenAlex ID: W2027153394
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description: scientific article; zbMATH DE number 3873528