Hom and Ext, Revisited

TL;DR

This paper investigates conditions under which modules over a Noetherian local ring are close to free, based on properties of Hom and Ext modules, providing a unified and simplified approach to existing results.

Contribution

It offers a unified, elementary method to extend and simplify known results relating Hom and Ext vanishing to module freeness over local rings.

Findings

Modules with certain Hom properties and Ext vanishing are close to free

Unified elementary approach to existing theorems

Extensions and simplifications of prior results in the literature

Abstract

Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated $R$-modules. We prove a number of results of the form: if $\mbox{Hom}_R(M,N)$ has some nice properties and $\mbox{Ext}^{1 \leq i \leq n}_R(M,N)=0$ for some $n$, then $M$ (and sometimes $N$) must be be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.