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stable general linear group

Let RR be an associative ring with 1. As usual GL n(R)GL_n(R) will denote the general linear group of n×nn\times n non-singular matrices over RR. There is an embedding of GL n(R)GL_n(R) into GL n+1(R)GL_{n+1}(R) sending a matrix M=(m i,j)M = (m_{i,j}) to the matrix M M^\prime obtained from MM by adding an extra row and column of zeros except that m n+1,n+1 =1m^\prime_{n+1,n+1} = 1. This gives a nested sequence of groups

GL 1(R)GL 2(R)GL n(R)GL n+1(R)GL_1(R)\subset GL_2(R)\subset \ldots \subset GL_n(R)\subset GL_{n+1}(R)\subset \ldots

and we write GL(R)GL(R) for the colimit (union in this case) of these. It will be called the stable general linear group over RR.

Created on February 3, 2012 14:32:34 by Tim Porter (95.147.237.181)