Absence of torsion in orbit space

Abstract

In this paper, we prove that if $R$ is a local ring of dimension $d,$ $d\ge 2$ and $\frac{1}{d!}\in R$ then the group $\frac{U{m}_{d+1}(R[X])}{E_{d+1}(R[X])}$ has no $k$-torsion, provided $k\in G{L}_1(R).$ We also prove that if $R$ is a regular ring of dimension $d,$ $d\ge 2$ and $\frac{1}{d!}\in R$ such that $E_{d+1}(R)$ acts transitively on $U{m}_{d+1}(R)$ then $E_{d+1}(R[X])$ acts transitively on $U{m}_{d+1}(R[X]).$