A nice group structure on the orbit space of unimodular rows-II

TL;DR

This paper proves a theorem about the group structure on the orbit space of unimodular rows, extending previous results to relative cases and specific dimensions, using an Excision type approach.

Contribution

It introduces an Excision type theorem for the niceness of the group structure on unimodular row orbit spaces, including relative cases and specific dimension scenarios.

Findings

Established niceness for the case n=d over smooth affine algebras

Extended results to relative versions for n=d+1

Proved niceness of the group structure using Excision techniques

Abstract

We establish an Excision type theorem for niceness of group structure on the orbit space of unimodular rows of length $n$ modulo elementary action. This permits us to establish niceness for relative versions of results for the cases when $n = d+1$, $d$ being the dimension of the base algebra. We then study and establish niceness for the case when $n = d$, and also establish a relative version, when the base ring is a smooth affine algebra over an algebraically closed field.