Systems of Parameters and the Cohen-Macaulay Property

TL;DR

This paper investigates the structure of homomorphism modules between quotients of a module over a local ring, extending known results from Cohen-Macaulay rings to more general modules.

Contribution

It generalizes Rees's isomorphism result for homomorphism modules from Cohen-Macaulay rings to arbitrary modules over local rings.

Findings

Hom modules exhibit new structural properties in non-Cohen-Macaulay contexts.

Extension of Rees's theorem to broader classes of modules.

Insights into the Cohen-Macaulay property via module homomorphisms.

Abstract

Let $R$ be a commutative, Noetherian, local ring and $M$ an $R$-module. Consider the module of homomorphisms $\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b} M)$ where $\mathfrak{b}\subseteq\mathfrak{a}$ are parameter ideals of $M$. When $M=R$ and $R$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is always isomorphic to $R/\mathfrak{a}$, and in particular, a free module over $R/\mathfrak{a}$ of rank one. In this work, we study the structure of such modules of homomorphisms for general $M$.