Hartshorne's questions and weakly cofiniteness

TL;DR

This paper investigates the properties of weakly Laskerian modules and their relation to Hartshorne's questions, establishing conditions under which local cohomology modules are weakly cofinite and the category of such modules is Abelian.

Contribution

It provides new criteria for weak cofiniteness of local cohomology modules and shows that the category of weakly cofinite modules with certain finiteness conditions is Abelian.

Findings

Under specified conditions, local cohomology modules are $a$-weakly cofinite.

The category of $a$-weakly cofinite $FD_{ ext{leq} 1}$ modules is Abelian.

Weakly Laskerian Ext modules are preserved under certain conditions.

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. The main purpose of this paper is to answer the Hartshorn's questions in the class of weakly Laskerian modules. It is shown that if $s\geq 1$ is a positive integer such that $\Ext^j_R(R/\fa, M)$ is weakly Laskerian for all $j\leq s$ and the $R$-module $H^i_\fa(M)$ is $FD_{\leq 1}$ for all $i < s$, then the $R$-module $H^i_\fa(M)$ is $\fa$-weakly cofinite for all $i <s$. In addition, we show that the category of all $\fa$-weakly cofinite $FD_{\leq 1}$ $R$-modules is an Abelian subcategory of the category of all $R$-modules. Also, we prove that if $\Ext^i_R(R/\fa,M)$ is weakly Laskerian for all $i\leq \dim M$, then the $R$-module $\Ext^i_R(N,M)$ is weakly Laskerian for all $i\geq 0$ and for any finitely generated $R$-module $N$ with $\Supp_R(N) \subseteq V (\fa)$ and $\dim N \leq 1$.