Bass numbers and endomorphism rings of Gorenstein injective modules

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 $G(R\mathrm{/}\mathfrak{p})$ is an $R_{\mathfrak{p}}$-module. We give some necessary and sufficient conditions under which $G(R\mathrm{/}\mathfrak{p})$ is an $R_{\mathfrak{p}}$-module. We also study the Bass numbers of $G(R\mathrm{/}\mathfrak{p})$ and we show that if ${\mathrm{G}\mathrm{i}\mathrm{d}}_{R}R\mathrm{/}\mathfrak{p}$ is finite, then ${\mu }^i(\mathfrak{q},G(R\mathrm{/}\mathfrak{p}))$ is finite for all $i\ge 0$ and all $\mathfrak{q}\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R$. If ${\mathrm{G}\mathrm{p}\mathrm{d}}_{R}R\mathrm{/}\mathfrak{p}$ is finite, then ${\mu }^i(\mathfrak{p},G(R\mathrm{/}\mathfrak{p}))$ is finite for all $i\ge 0$. We define a subring $S(\mathfrak{p}{)}_{\mathfrak{p}}$ of ${\mathrm{E}\mathrm{n}\mathrm{d}}_{R_{\mathfrak{p}}}(G(R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}))$ and we show that it is noetherian and contains a subring which is a quotient of $\widehat{R_{\mathfrak{p}}}$.