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Gorenstein injectivity of the section functor

Let $R$ be a commutative Noetherian ring of Krull dimension $d$ admitting a dualizing complex $D$ and let $\frak a$ be any ideal of $R$, we prove that $Γ_{\frak a}(G)$ is Gorenstein injective for any Gorenstein injective $R$-module $G$. Let $(R,\frak m)$ be a local ring and $M$ be a finitely generated $R$-module. We show that ${\rm Gid}{\bf R}Γ_{\frak m}(M)<\infty$ if and only if ${\rm Gid}_{\hat{R}}(M\otimes_R\hat{R})<\infty$. We also show that if ${\rm Gfd}_R{\bf R}Γ_{\frak m}(M)<\infty$, then ${\rm Gfd}_RM<\infty$. Let $(R,\frak m)$ be a Cohen-Macaulay local ring and $M$ be a Cohen-Macaulay module of dimension $n$. We prove that if $H_{\frak m}^n(M)$ is of finite G-injective dimension, then Gid$_RH_{\frak m}^n(M)=d-n$. Moreover, we prove that if $M$ is a Matlis reflexive strongly torsion free module of finite G-flat dimension, then Gfd$_R\hat{M}<\infty$, where $\hat{M}$ is $\frak m$-adic completion.