Modules cofinite with respect to ideals of small dimensions

TL;DR

This paper characterizes modules cofinite with respect to ideals of small dimension in noetherian rings, extending previous results and providing new insights into the cofiniteness of local cohomology modules.

Contribution

It generalizes a key result on $rak{a}$-cofiniteness for modules of dimension at most 2 and introduces new findings on the cofiniteness of local cohomology modules.

Findings

Modules of dimension ≤ 2 are $rak{a}$-cofinite iff certain Ext modules are finitely generated.

Provides new criteria for cofiniteness of local cohomology modules.

Extends existing theorems to broader classes of modules and ideals.

Abstract

Let $\mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $\mathrm{Supp}_RM\subseteq\mathrm{V}(\mathfrak{a})$. We show that if $\mathrm{dim}_RM\leq2$, then $M$ is $\mathfrak{a}$-cofinite if and only if $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ are finitely generated for all $i\leq 2$, which generalizes one of the main results in [Algebr. Represent. Theory 18 (2015) 369--379]. Some new results concerning cofiniteness of local cohomology modules $\mathrm{H}^i_\mathfrak{a}(M)$ for any finitely generated $R$-module $M$ are obtained.