Cofiniteness properties of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. Let $t\in\Bbb{N}_0$ be an integer, $M$ a finitely generated $R$-module and $X$ be an $R$-module such that $\Ext^i_R(R/I,X)$ is finitely generated (resp. minimax, weakly Laskerian) for all $i\leq t+1$. We prove that if $H^{i}_{I}(M, X)$ is ${\rm FD_{\leq 1}}$ for all $i<t$, then the $R$-modules $H^{i}_{I}(M,X)$ are $I$-cofinite (resp. $I$-cominimax, $I$-weakly cofinite) for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $H^{t}_{I}(M,X)$, the $R$-module $\Ext^i_R(R/I,H^{t}_{I}(M,X)/N)$ is finitely generated (resp. minimax, weakly Laskerian) for $i=0,1$. In particular the set $Ass_R(H^{t}_{I}(M,X)/N)$ is a finite set.