Cofiniteness of local cohomology modules in the class of modules in dimension less than a fixed integer

Abstract

‎Let $n$ be a non-negative integer‎, ‎$R$ a commutative Noetherian ring with $\dim(R)\leq n‎+ ‎2$‎, ‎$\mathfrak{a}$ an ideal of $R$‎, ‎and $X$ an arbitrary $R$-module‎. ‎In this paper‎, ‎we first prove that $X$ is an $(\operatorname{FD}_{< n}‎, ‎\mathfrak{a})$-cofinite $R$-module if $X$ is an $\mathfrak{a}$-torsion $R$-module such that $\operatorname{Hom}_{R}(\frac{R}{\mathfrak{a}}‎, ‎X)$ and $\operatorname{Ext}_{R}^{1}(\frac{R}{\mathfrak{a}}‎, ‎X)$ are $\operatorname{FD}_{< n}$ $R$-modules‎. ‎Then‎, ‎we show that $\operatorname{H}^{i}_{\mathfrak{a}}(X)$ is an $(\operatorname{FD}_{< n}‎, ‎\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in \operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X))‎ : ‎\dim(R/\mathfrak{p})\geq n\}$ is a finite set for all $i$ when $\operatorname{Ext}_{R}^{i}(\frac{R}{\mathfrak{a}}‎, ‎X)$ is an $\operatorname{FD}_{< n}$ $R$-module for all $i\leq n‎+ ‎2$‎. ‎As a consequence‎, ‎it follows that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X))$ is a finite set for all $i$ whenever $R$ is a semi-local ring with $\dim(R)\leq 3$ and $X$ is an $\operatorname{FD}_{< 1}$ $R$-module‎. ‎Finally‎, ‎we observe that the category of $(\operatorname{FD}_{< n}‎, ‎\mathfrak{a})$-cofinite $R$-modules forms an Abelian subcategory of the category of $R$-modules.