Gorenstein injectivity of the section functor

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Let R be a commutative Noetherian ring of Krull dimension d admitting a dualizing complex D and let a be any ideal of R, we prove that Γ a(G) is Gorenstein injective for any Gorenstein injective R-module G. Let (R, m) be a local ring and M be a finitely generated R-module. We show that Gid RΓ m(M)<∞ if and only if GidR(M⊗ᵣR)<∞. We also show that if Gfdᵣ RΓ m(M)<∞, then GfdᵣM<∞. Let (R, m) be a Cohen-Macaulay local ring and M be a Cohen-Macaulay module of dimension n. We prove that if H mⁿ(M) is of finite G-injective dimension, then GidᵣH mⁿ(M)=d-n. Moreover, we prove that if M is a Matlis reflexive strongly torsion free module of finite G-flat dimension, then GfdᵣM<∞, where M is m-adic completion.