On the cofiniteness of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring, let $I$ be an ideal of $R$, and let $M$, $N$ be two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $H_I^j(M,N)={\underset{⟶}{\lim }}_n{Ext}_R^j(M/I^{n}M,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal, then $H_I^j(M,N)$ is $I$-cofinite for all $M$, $N$ and all $j$. Secondly, let $t$ be a nonnegative integer such that $dimSupp\left(H_I^j\right(M,N\left)\right)\le 1$ for all $j<t$. Then $H_I^j(M,N)$ is $I$-cofinite for all $j<t$ and $Hom(R/I,H_I^t(M,N\left)\right)$ is finitely generated. Finally, we show that if $dim\left(M\right)\le 2$ or $dim\left(N\right)\le 2$, then $H_I^j(M,N)$ is $I$-cofinite for all $j$.