On the relative Cohen-Macaulay modules

TL;DR

This paper studies modules over commutative Noetherian local rings that are relative Cohen-Macaulay with respect to an ideal, showing that for Gorenstein modules, the local cohomology is concentrated in one degree and characterized by homological properties.

Contribution

It generalizes previous results by demonstrating that Gorenstein modules have local cohomology concentrated in a single degree, characterized by Bass numbers, extending known results for the ring itself.

Findings

Gorenstein modules have local cohomology concentrated in one degree.

The non-vanishing local cohomology module's properties determine the module's homological dimension.

Generalization of a known result for the ring to Gorenstein modules.

Abstract

Let $R$ be a commutative Noetherian local ring and let $\fa$ be a proper ideal of $R$. A non-zero finitely generated $R$-module $M$ is called relative Cohen-Macaulay with respect to $\fa$ if there is precisely one non vanishing local cohomology modules $\H_{\fa}^{i}(M)$ of $M$. In this paper, as a main result, it is shown that if $M$ is a Gorenstein $R$--module, then $\H_{\fa}^{i}(M)=0$ for all $i\neq c$ where $c=\h_{M}\fa$ is completely encoded in homological properties of $\H_{\fa}^{c}(M)$, in particular in its Bass numbers. Notice that, this result provides a generalization of a result of M. Hellus and P. Schenzel which has been proved before, as a main result, in the case where $M=R$.