Modules cofinite with respect to ideals of small dimensions

Abstract

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

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