Gorenstein injectivity of the section functor

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 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 if and only if . We also show that if , then Gfd R M < ∞. Let ( R , 𝔪) be a Cohen-Macaulay local ring and M be a Cohen-Macaulay module of dimension n . We prove that if is of finite G-injective dimension, then = d − 1. Moreover, we prove that if M is a Matlis reflexive strongly torsion free module of finite G-flat dimension, then , where is 𝔪-adic completion.