Weakly cofiniteness of local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $I\in \Phi$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\dim M\leq 1$, then the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all $i\geq 0$, if and only if the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for $i=0, 1$. Let $t\in\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $\Ext^i_R(R/I,M)$ is weakly Laskerian for all $i\leq t+1$. We prove that if the $R$-module $\lc^{i}_\Phi(M)$ is ${\rm FD_{\leq 1}}$ for all $i<t$, then $\lc^{i}_\Phi(M)$ is $\Phi$-weakly cofinite for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $\lc^t_\Phi(M)$, the $R$-modules $\Hom_R(R/I,\lc^t_\Phi(M)/N)$ and $\Ext^1_R(R/I,\lc^t_\Phi(M)/N)$ are weakly Laskerian. Let $N$ be a finitely generated $R$-module. We also prove that $\Ext^j_R(N,\lc^{i}_\Phi(M))$ and ${\rm Tor}^R_{j}(N,H^{i}_\Phi(M))$ are $\Phi$-weakly cofinite for all $i$ and $j$ whenever $M$ is weakly Laskerian and $\lc^{i}_\Phi(M)$ is ${\rm FD_{\leq 1}}$ for all $i$. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.