\(K_ 2\) of non-commutative local rings
type: Item
instance of: scholarly article
title: \(K_ 2\) of non-commutative local rings
zbMATH Open document ID: 0588.16019
DOI: 10.1016/0021-8693(85)90100-0
author: Manfred Kolster
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: A presentation for \(K_ 2\) of commutative local rings with finite residue fields using Dennis-Stein symbols is well-known [eg. see \textit{H. Maazen} and \textit{J. Stienstra}, J. Pure Appl. Algebra 10, 271-294 (1977; Zbl 0393.18013)]. This paper proves a noncommutative analog which is similar to the \textit{R. K. Dennis} and \textit{M. R. Stein} presentation for \(K_ 2\) of a discrete valuation ring [Adv. Math. 18, 182-238 (1975; Zbl 0318.13017)]. The author sets \(D_ 1(A)\) to be the group generated by Dennis-Stein symbols, i.e. \(<a,b>\) such that \(1+ab\in A^*\) (the group of units of A), and satisfying 4 relations. The relations are a little different from the usual commutative ones and are used to show that the two most complicated relations for a discrete valuation ring given by Dennis and Stein [loc. cit] are redundant. The main result in this paper is that \[ 0\to K_ 2(A)\to D_ 1(A)\to [A^*,A^*]\to 0 \] is exact for any local ring. If A is commutative, the sequence reduces to the usual presentation for \(K_ 2(A)\).
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 3944070
zbMATH Keywords: presentation for \(K_ 2\)
zbMATH Keywords: local rings
zbMATH Keywords: Dennis-Stein symbols
zbMATH Keywords: relations
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90100-0
OpenAlex ID: W1995165807
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description: scientific article; zbMATH DE number 3944070