Weakly cofiniteness of local cohomology modules

TL;DR

This paper investigates the weak cofiniteness properties of local cohomology modules over Noetherian rings, establishing conditions under which these modules are weakly Laskerian and cofinite, and exploring their structural properties.

Contribution

It provides new criteria for weak cofiniteness of local cohomology modules, extending known results to modules with certain finiteness conditions and generalizing to modules defined by pairs of ideals.

Findings

Weakly Laskerian property of Ext modules for modules with dim ≤ 1.

Conditions ensuring local cohomology modules are weakly cofinite.

Results on cofiniteness of Ext and Tor modules involving local cohomology.

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.