A nice group structure on the orbit space of unimodular rows. II (Q402673)

This article is the second part of [\textit{A. S. Garge} and \textit{R. A. Rao}, \(K\)-Theory 38, No. 2, 113--133 (2008; Zbl 1143.13013)]. Let \(R\) be a ring with stable dimension \(d\) and \(I\) be any ideal in it. \(\mathrm{Um}_n(R,I)\) denotes the set of unimodular rows of length \(n\) in \(R\) such that the first \(n-1\) entries are from \(I\) and the last entry is comaximal with \(I\). \(E_n(R,I)\) denotes the normal closure of the subgroup of elementary matrices generated by \(E_{ij}(a)\) such that \(a \in I\).NEWLINENEWLINEFrom results of \textit{W. van der Kallen} [J. Algebra 82, 363--397 (1983; Zbl 0518.20035); J. Pure Appl. Algebra 57, No. 3, 281--316 (1989; Zbl 0665.18011)], it is known that \(\mathrm{MSE}_n(R, I) = \frac{\mathrm{Um}_n(R)}{E_n(R)}\) has a group structure when \(n \geq 3\) and \(d \leq 2n - 4\). The authors refer of the paper under consideration refer to the multiplication rule as the Vv rule.NEWLINENEWLINEIn these articles, the authors consider the question of when this multiplication is ``nice'', i.e. when is the multiplication given by NEWLINE\[NEWLINE[a_1, a_2, \ldots, a_{n-1}, a] * [a_1, a_2, \ldots, a_{n-1}, b] = [a_1, a_2, \ldots, a_{n-1}, ab] ?NEWLINE\]NEWLINE In the previous article, the authors gave affirmative answers for this question in the non-relative case when \(n= d+1\) with some hypotheses on the ring \(R\). In this article, the authors first prove that even in the relative case as above, multiplication does not depend on the choice of coordinate and subsequently give an affirmative answer to the above question in the following cases: {\parindent=6mm \begin{itemize}\item[1.] \(R\) is a smooth affine algebra over an algebraically closed field \(k, \text{ char.}(k) \neq 2,3,n=d=3\) and \(I\) is a principal ideal, \newline \item[2.] \(R\) is a smooth affine algebra over an algebraically closed field \(k, \text{ char}(k) \neq 2,3,n=d \geq 4\). NEWLINENEWLINE\end{itemize}} Another interesting result is a relative version of the Mennicke-Newman theorem. The proofs are in the same spirit as the previous article with a few key new ingredients coming from \(\mathbb{A}^1-\)homotopy theory and a theorem of J. Fasel about when stably elementary matrices are stably elementary symplectic.