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

摘要

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