On \(K_2\)-group of a formal matrix ring
type: Item
instance of: scholarly article
title: On \(K_2\)-group of a formal matrix ring
zbMATH Open document ID: 1266.19001
author: Ziqiang Fan
author: Zhixiang Yin
publication date: 2013-03-15T00:00:00Z
review text: The authors give a presentation of Milnor's \(K_2\) group of a formal matrix ring, under the assumption that the off-diagonal entries form an ideal. More precisely, assume (for \(i,j=1,\dots,t\)) that \(S_{ii}\) are rings and \(S_{ij}\) are \(S_{ii},S_{jj}\) bimodules endowed with composition maps \(S_{ij} \otimes S_{jk} \rightarrow S_{ik}\) that make \(S = \sum S_{ij}\) an associative ring. The typical example would be a semiperfect ring. Assume further that \(S_{ij}S_{ji} = 0\), so that \(I = \sum_{i\neq j} S_{ij}\) is a split radical ideal. Then the short exact sequence \[ 0 \longrightarrow K_2(R,I) \longrightarrow K_2(R) \longrightarrow K_2(R/I) \longrightarrow 0 \] splits. In what follows the notation \(x_{\alpha\beta}\) will always refer to an element of \(S_{\alpha\beta}\). The authors show that \(K_2(R,I)\) is generated by the symbols \(\langle a_{ij},b_{ji} \rangle\) (\(i\neq j\)), subject to the following sets of relations: antisymmetry, biadditivity, transfer of scalars (i.e., \(\langle a_{ij}r_{jj},b_{ji} \rangle = \langle a_{ij},r_{jj}b_{ji} \rangle\)) and \(\langle a_{ij},b_{jk}c_{ki} \rangle \langle b_{jk},c_{ki}a_{ij} \rangle \langle c_{ki},a_{ij}b_{jk}\rangle=1\) for distinct \(i,j,k\).
reviewed by: Uzi Vishne
Mathematics Subject Classification ID: 19C20
Mathematics Subject Classification ID: 16L99
zbMATH DE Number: 6145423
zbMATH Keywords: \(K_2\)-group
zbMATH Keywords: formal matrix ring
MaRDI profile type: Publication
full work available at URL: s10114-012-9466-y
OpenAlex ID: W2395343874
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DOI: 10.1007/S10114-012-9466-Y
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6145423