Cofinite modules and cofiniteness of local cohomology modules

Abstract

Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$-module, and $X$ an arbitrary $R$-module. In this paper, we first prove that if $\dim_R(M)\leq{n+2}$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M)):\dim(R/\mathfrak{p})\geq{n}\}$ is a finite set for all $i$. As a consequence, it follows that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M))$ is a finite set for all $i$ when $R$ is a semi-local ring and $\dim_R(M)\leq{3}$. Then, we show that if $\dim(R/\mathfrak{a})\leq{n+1}$, then $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i$ whenever $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i\leq{\dim_R(X)-n}$. Finally, in the case that $\dim(R/\mathfrak{a})\leq{2}$, $X$ is $\mathfrak{a}$-torsion, and $n>0$ or $\operatorname{Supp}_R(X)\cap\operatorname{Var}(\mathfrak{a})\cap\operatorname{Max}(R)$ is finite, we prove that $X$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$-cofinite $R$-module when $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i\leq{2-n}$. We conclude with some ordinary $\mathfrak{a}$-cofiniteness results for local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(X)$.