Absence of torsion in orbit space

TL;DR

This paper proves the absence of torsion in certain orbit groups over local rings and establishes transitivity of elementary groups on unimodular rows over polynomial extensions.

Contribution

It extends known results by showing torsion-free properties and transitivity of elementary groups in the context of polynomial rings over specific local and regular rings.

Findings

The group +1(R[X]) / E_{d+1}(R[X]) has no k-torsion for certain rings.

E_{d+1}(R[X]) acts transitively on unimodular rows under specified conditions.

Results apply to local rings of dimension  with 1/d! in R and regular rings of dimension .

Abstract

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