Bass numbers and endomorphism rings of Gorenstein injective modules

Abstract

Let $R$ be a commutative noetherian ring admitting a dualizing complex and let $\mathfrak p$ be a prime ideal of $R$. In this paper we investigate when $G(R/\frak p)$ is an $R_{\frak p}$-module. We give some necessary and sufficient conditions under which $G(R/\frak p)$ is an $R_{\frak p}$-module. We also study the Bass numbers of $G(R/\frak p)$ and we show that if ${\rm Gid}_RR/\frak p$ is finite, then $\mu^i(\frak q,G(R/\frak p))$ is finite for all $i\geq 0$ and all $\frak q\in{\rm Spec} R$. If ${\rm Gpd}_RR/\frak p$ is finite, then $\mu^i(\frak p,G(R/\frak p))$ is finite for all $i\geq 0$. We define a subring $S(\frak p)_{\frak p}$ of ${\rm End}_{R_{\frak p}}(G(R_{\frak p}/\frak pR_{\frak p}))$ and we show that it is noetherian and contains a subring which is a quotient of $\widehat{R_{\frak p}}$.