Gorenstein injective modules and local cohomology

Abstract:

In this paper we assume that $R$ is a Gorenstein Noetherian ring. We show that if $(R,\mathfrak {m})$ is also a local ring with Krull dimension $d$ that is less than or equal to 2, then for any nonzero ideal $\mathfrak {a}$ of $R$ , $H_{\mathfrak {a}}^d(R)$ is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if $R$ is a Gorenstein ring, then for any $R$-module $M$ its local cohomology modules can be calculated by means of a resolution of $M$ by Gorenstein injective modules. Also we prove that if $R$ is $d$-Gorenstein, $M$ is a Gorenstein injective and $\mathfrak a$ is a nonzero ideal of $R$, then ${\Gamma }_{\mathfrak {a}}(M)$ is Gorenstein injective.