Cofiniteness of local cohomology modules over Noetherian local rings

Abstract

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module of dimension $d\geq 1$. In this paper we consider the $I$-cofiniteness property of the local cohomology module $H^{d-1}_I(M)$. More precisely, we prove that the $R$-module $H^{d-1}_I(M)$ is $I$-cofinite if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(M))$ is finitely generated. As an immediate consequence of this result, we prove that if $(R,\operatorname{\frak m})$ is a regular local ring of dimension $d\geq 2$ and $I$ is an ideal of $R$ with $\dim R/I\neq 1$, then $H^{d-1}_I(R)=0$ if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(R))$ is finitely generated.