A note on cancellation of projective modules

TL;DR

This paper investigates conditions under which projective modules over a ring are cancellative, extending known results by analyzing the action of elementary groups on unimodular elements in the context of ring extensions.

Contribution

It establishes a new criterion involving elementary groups acting transitively on unimodular elements, generalizing previous results by Gubeladze for projective modules over rings of dimension d.

Findings

E_{d+1}(R) acts transitively on Um_{d+1}(R) for finite extensions R of A

E(A⊕P) acts transitively on Um(A⊕P) for projective modules P of rank d

Generalization of Gubeladze's results on projective modules and cancellation

Abstract

Let $A$ be a ring of dimension $d$. Assume that for every finite extension ring $R$ of $A$, E_{d+1}(R) acts transitively on Um_{d+1}(R). Then we prove that E(A\oplus P) acts transitively on Um(A\oplus P), for any projective A-module P of rank d. As a consequence of this, we generalise some results of Gubeladze.