Systems of parameters and the Cohen-Macaulay property (Q5916064)

Let \(R\) be a local commutative Noetherian ring, \(M\) be a finitely generated \(R\)-module, and \(\mathfrak b\subseteq \mathfrak a\) be pararmeter ideals of \(M\). This paper gives many situations in which the \(R/\mathfrak a\) module \(\text{Hom}_R(R/\mathfrak a,M/\mathfrak b M)\) is decomposable and not free. These situations stand in contrast to the situation where \(R=M\) is Cohen-Macaulay and \(\text{Hom}_R(R/\mathfrak a,M/\mathfrak b M)\) is isomorphic to \(R/\mathfrak a\).