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 $\dim M\le 1$, then the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all $i\ge 0$, if and only if the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for $i=0,1$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that $\Ext^i_R(R/I,M)$ is weakly Laskerian for all $i\le t+1$. We prove that if the $R$-module $\lc^{i}_Φ(M)$ is $\mathrm{F}{\mathrm{D}}_{\le 1}$ for all $i<t$, then $\lc^{i}_Φ(M)$ is $\Phi$-weakly cofinite for all $i<t$ and for any $\mathrm{F}{\mathrm{D}}_{\le 0}$ (or minimax) submodule $N$ of $\lc^t_Φ(M)$, the $R$-modules $\Hom_R(R/I,\lc^t_Φ(M)/N)$ and $\Ext^1_R(R/I,\lc^t_Φ(M)/N)$ are weakly Laskerian. Let $N$ be a finitely generated $R$-module. We also prove that $\Ext^j_R(N,\lc^{i}_Φ(M))$ and ${\mathrm{Tor}}_j^R(N,H_{\Phi }^i(M))$ are $\Phi$-weakly cofinite for all $i$ and $j$ whenever $M$ is weakly Laskerian and $\lc^{i}_Φ(M)$ is $\mathrm{F}{\mathrm{D}}_{\le 1}$ for all $i$. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.

16 pages