The Quotient Unimodular Vector Group is Nilpotent

Abstract

Jose–Rao introduced and studied the Special Unimodular Vector group $\mathrm{SUm_r(R)}$ and $\mathrm{EUm_r(R)}$, its Elementary Unimodular Vector subgroup. They proved that for $r \ge 2$, $\mathrm{EUm_r(R)}$ is a normal subgroup of $\mathrm{SUm_r(R)}$. The Jose–Rao theorem says that the quotient Unimodular Vector group, $\mathrm{SUm_r(R)}/\mathrm{EUm_r(R)}$, for $r \ge 2$, is a subgroup of the orthogonal quotient group $\mathrm{SO}_{2(r+1)}(R)/{\mathrm{EO}}_{2(r + 1)}(R)$. The latter group is known to be nilpotent by the work of Hazrat–Vavilov, following methods of A. Bak; and so is the former. In this article we give a direct proof, following ideas of A. Bak, to show that the quotient Unimodular Vector group is nilpotent of class $\le d = \dim (R)$. We also use the Quillen–Suslin theory, inspired by A. Bak’s method, to prove that if $R = A[X]$, with A a local ring, then the quotient Unimodular Vector group is abelian.