The quotient Unimodular Vector group is nilpotent

Reema Khanna; Selby Jose; Sampat Sharma; Ravi A. Rao

doi:10.1007/978-981-15-1611-5_13

paper

The quotient Unimodular Vector group is nilpotent

arXiv:2001.06976 · doi:10.1007/978-981-15-1611-5_13

Abstract

Jose-Rao introduced and studied the Special Unimodular Vector group $S U m_{r} (R)$ and $E U m_{r} (R)$ , its Elementary Unimodular Vector subgroup. They proved that for $r \geq 2$ , $E U m_{r} (R)$ is a normal subgroup of $S U m_{r} (R)$ . The Jose-Rao theorem says that the quotient Unimodular Vector group, $S U m_{r} (R) / E U m_{r} (R)$ , for $r \geq 2$ , is a subgroup of the orthogonal quotient group $S O_{2 (r + 1)} (R) / E O_{2 (r + 1)} (R)$ . 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 $\leq d = dim (R)$ . 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.