Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

TL;DR

This paper investigates the properties of local cohomology modules with respect to a semidualizing module in a local ring, providing new characterizations of dualizing and Gorenstein modules based on injective dimensions.

Contribution

It introduces new relationships between $C$-injective and $G_C$-injective dimensions of local cohomology modules and characterizes Gorenstein rings via these properties.

Findings

Comparison of injective dimensions of $C$ and local cohomology modules

Characterization of dualizing modules using $C$-injectivity

Gorenstein property of $R$ when certain local cohomology modules are $C$-injective

Abstract

Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently presented.