An Abelian group structure on orbits of ``unimodular squares'' in dimension 3
Let \(A\) be a commutative noetherian ring of dimension \(d\) and \(n\geq 3\). Consider the orbit set \(Um_n(A)/E_n(A)\), where \(E_n(A)\) is generated by elementary matrices and \(Um_n(A)\) consists of the rows of length \(n\) whose entries generate the unit ideal in \(A\). In the case \(d=2\), \(n=3\) it was discovered by Vaserstein that \(Um_3(A)/E_3(A)\) carries a natural (abelian) group structure [\textit{L. N. Vaserstein} and \textit{A. A. Suslin}, Izv. Akad. Nauk SSSR, Ser. Mat. 40, 993-1054 (1976; Zbl 0338.13015)]. This result was extended by the reviewer to the range \(n\geq\max(3,\tfrac 12d+2)\), which naturally arises in algebraic topology (in the suspension theorem) [see J. Pure Appl. Algebra 57, No. 3, 281-316 (1989; Zbl 0665.18011)]. The author now considers the case \(d=n=3\), which is just outside the range. Instead of putting a group structure on all of \(Um_3(A)/E_3(A)\), he puts one on \(\psi_2(Um_3(A))/E_3(A)\), where \(\psi_2\) is taken from [\textit{L. N. Vaserstein}; J. Algebra 100, 456-461 (1986; Zbl 0592.20053)]. This \(\psi_2\) squares the first entry of a row. Rao exploits the completion given by \textit{M. Krusemeyer} of an element of \(\psi_2(Um_3(A))\) to a 3 by 3 matrix of determinant 1 [Commun. Algebra 4, 657-663 (1976; Zbl 0348.15009)]. This he mixes with Vaserstein's original construction.
scientific article; zbMATH DE number 1247574