Stability results for projective modules over Rees algebras

TL;DR

This paper establishes stability and cancellation properties for projective modules over certain Rees algebras, providing new insights into their structure and applications to module generation.

Contribution

It introduces a class of Noetherian domains where projective modules of rank equal to the dimension split off a free summand and are cancellative.

Findings

Projective modules of rank d split off a free summand of rank one.

Modules over these Rees algebras are cancellative.

Applications to the number of generators of modules.

Abstract

We provide a class of commutative Noetherian domains $R$ of dimension $d$ such that every finitely generated projective $R$-module $P$ of rank $d$ splits off a free summand of rank one. On this class, we also show that $P$ is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.