Cofiniteness with Respect to Two Ideals and Local Cohomology

Abstract

Let A be a noetherian ring and $\frak a$ be an ideal of A. We define a condition P$_{n}(\frak a)$ for $\frak a$-cofiniteness of modules and we show that if A is of dimension d satisfying P$_{d-1}(\frak a)$ for all ideals of dimension d − 1, then it satisfies P$_{d-1}(\frak a)$ for all ideals $\frak a$. Let M be an A-module and let n be a non-negative integer such that ${\text {Ext}_{A}^{i}}(A/\frak a,M)$ is finite for all i ≤ n + 1. We show that if $\dim \mathrm {A}/\frak a = 1$, then $H_{\frak a}^{i}(M)$ is $\frak a$-cofinite for all i ≤ n and if A is local with $\dim \mathrm {A}/\frak a = 2$, then $H_{\frak a}^{i}(M)$ is $\frak a$-cofinite for all i < n if and only if $\text {Hom}_{\mathrm {A}}(\mathrm {A}/\frak a,\mathrm {H}_{\frak a}^{i}(\mathrm {M}))$ is finite for all i ≤ n. Finally we prove that if M is an A-module of dimension d such that $(0:_{H_{\frak a}^{d}(M)}\frak a)$ is finite, then $H_{\frak a}^{d}(M)$ is artinian.