On the existence of unimodular elements and cancellation of projective modules over noetherian and non-noetherian rings

TL;DR

This paper investigates conditions under which projective modules over certain rings have unimodular elements and are cancellative, extending classical results to non-noetherian rings and rings over Prüfer domains.

Contribution

It establishes new criteria for the existence of unimodular elements and cancellation of projective modules over both noetherian and non-noetherian rings, including rings of finite type over Prüfer domains.

Findings

Projective modules of rank ≥ d+1 are cancellative if they have a unimodular element.

Modules of rank ≥ dim(S) have unimodular elements, ensuring cancellation.

Results extend classical theorems to broader classes of rings, including non-noetherian cases.

Abstract

Let $R$ be a commutative ring of dimension $d$, $S = R[X]$ or $R[X, 1/X]$ and $P$ a finitely generated projective $S$ module of rank $r$. Then $P$ is cancellative if $P$ has a unimodular element and $r \geq d + 1$. Moreover if $r \geq \dim (S)$ then $P$ has a unimodular element and therefore $P$ is cancellative. As an application we have proved that if $R$ is a ring of dimension $d$ of finite type over a Pr\"{u}fer domain and $P$ is a projective $R[X]$ or $R[X, 1/X]$ module of rank at least $d + 1$, then $P$ has a unimodular element and is cancellative.