GE2-rings and a graph of unimodular rows |
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Institution: | School of Mathematics and Statistics, University College Dublin, Ireland |
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Abstract: | For a commutative ring A we consider a related graph, , whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that is path-connected if and only if A is a -ring, in the terminology of P. M. Cohn. Furthermore, if denotes the clique complex of , we prove that is simply connected if and only if A is universal for . More precisely, our main theorem is that for any commutative ring A the fundamental group of is isomorphic to the group modulo the subgroup generated by symbols. |
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Keywords: | Special linear group Elementary matrices |
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