Cofiniteness of local cohomology modules for ideals of dimension one

TL;DR

This paper investigates the conditions under which local cohomology modules are cofinite or weakly Laskerian, establishing equivalences with finiteness properties of certain Ext modules for ideals of dimension one.

Contribution

It generalizes previous results by providing new criteria linking cohomology module cofiniteness with Ext module finiteness for ideals of dimension one.

Findings

Equivalence between Ext module finiteness and local cohomology cofiniteness for dimension one ideals.

Extension of known results to broader classes of modules and ideals.

Conditions ensuring cofiniteness of local cohomology modules in terms of Ext modules.

Abstract

Let $R$ denote a commutative Noetherian (not necessarily local) ring, $M$ an arbitrary $R$-module and $I$ an ideal of $R$ of dimension one. It is shown that the $R$-module $\Ext^i_R(R/I,M)$ is finitely generated (resp. weakly Laskerian) for all $i\leq {\rm cd}(I,M)+1$ if and only if the local cohomology module $H^i_I(M)$ is $I$-cofinite (resp. $I$-weakly cofinite) for all $i$. Also, we show that when $I$ is an arbitrary ideal and $M$ is finitely generated module such that the $R$-module $H^i_I(M)$ is weakly Laskerian for all $i\leq t-1$, then $H^i_I(M)$ is $I$-cofinite for all $i\leq t-1$ and for any minimax submodule $K$ of $H^{t}_I(M)$, the $R$-modules $\Hom_R(R/I, H^{t}_I(M)/K)$ and $\Ext^{1}_R(R/I, H^{t}_I(M)/K)$ are finitely generated, where $t$ is a non-negative integer. This generalizes the main result of Bahmanpour-Naghipour \cite{BN} and Brodmann and Lashgari \cite{BL}.