Elementary matrix-computational proof of Quillen-Suslin theorem for Ore extensions

TL;DR

This paper provides an elementary, matrix-based proof of the Quillen-Suslin theorem for Ore extensions, along with an algorithm to compute bases of finitely generated projective modules, implemented in a computational package.

Contribution

It introduces a new elementary proof and an algorithm for free modules over Ore extensions, with implementation and examples.

Findings

Algorithm successfully computes bases of projective modules.

Implementation demonstrates practical applicability.

Proof simplifies understanding of the Quillen-Suslin theorem for Ore extensions.

Abstract

In this short note we present an elementary matrix-constructive proof of Quillen-Suslin theorem for Ore extensions: If $K$ is a division ring and $A:=K[x;\sigma,\delta]$ is an Ore extension, with $\sigma$ bijective, then every finitely generated projective $A$-module is free. We will show an algorithm that computes the basis of a given finitely generated projective module. The algorithm has been implemented in a computational package, and some illustrative examples are included.