Homologie du groupe linéaire et K-théorie de Milnor des anneaux. (Homology of the linear group and Milnor's K-theory of rings)
type: Item
instance of: scholarly article
title: Homologie du groupe linéaire et K-théorie de Milnor des anneaux. (Homology of the linear group and Milnor's K-theory of rings)
zbMATH Open document ID: 0669.20037
DOI: 10.1016/0021-8693(89)90033-1
author: Daniel Guin
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let A be an associative ring with 1. \(GL_ nA\) the group of invertible n by n matrices over A. The general theorem of Suslin says that the canonical homomorphism \(H_ k(GL_ nA)\to H_ k(GL_{n+1}A)\) is surjective for \(n\geq \max (2k,k+sr(A)-1)\) and bijective for \(n\geq \max (2k+1,k+sr(A))\), where sr(A) is the stable rank of A. In the present paper it is shown that the above homomorphism is an isomorphism for \(n\geq k\) in the case when \(sr(A)=1\) and A has ``sufficiently many'' units (the conditions H1 and H2 of the paper). This allows us to identify the cokernel of this homomorphism with \(n=k-1\) with the Milnor group \(K^ M_ n(A)\). \textit{A. A. Suslin} and \textit{Yu. P. Nesterenko} obtained recently [Izv. Akad. Nauk SSSR, Ser. Mat. 53, No.1, 121-146 (1989)] these results for commutative local rings A with infinite residue fields. They also showed that if a commutative A satisfies H2, then the above bounds for stability can be replaced by \(n\geq k+sr(A)-1\) and \(n\geq k+sr(A)\) respectively.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 4095771
zbMATH Keywords: homomorphisms in homology
zbMATH Keywords: stable rank
zbMATH Keywords: isomorphism
zbMATH Keywords: Milnor group
zbMATH Keywords: bounds for stability
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90033-1
OpenAlex ID: W2037938151
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description: scientific article; zbMATH DE number 4095771