Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

TL;DR

This paper investigates the elementary action of classical groups on unimodular rows over monoid rings, establishing transitivity conditions and stabilization bounds that extend previous results for linear groups.

Contribution

It extends Gubeladze's results by analyzing symplectic and orthogonal groups' actions on unimodular rows over monoid rings, providing new transitivity and stabilization bounds.

Findings

Transitivity of symplectic and orthogonal groups on unimodular rows under certain conditions.

Establishment of surjective stabilization bounds for K_1 of classical groups.

Extension of Gubeladze's results from linear groups to symplectic and orthogonal groups.

Abstract

The elementary action of symplectic and orthogonal groups on unimodular rows of length $2n$ is transitive for $2n \geq \max(4, d+2)$ in the symplectic case, and $2n \geq \max(6, 2d+4)$ in the orthogonal case, over monoid rings $R[M]$, where $R$ is a commutative noetherian ring of dimension $d$, and $M$ is commutative cancellative torsion free monoid. As a consequence, one gets the surjective stabilization bound for the $K_1$ for classical groups. This is an extension of J. Gubeladze's results for linear groups.