Systems of parameters and the Cohen-Macaulay property

Abstract

Let $R$ be a commutative, Noetherian, local ring and $M$ a finitely generated $R$-module. Consider the module of homomorphisms $Hom _R(R/\mathfrak{a} ,M/\mathfrak{b} M)$ 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 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 a not necessarily Cohen-Macaulay $R$-module $M$.