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1936-1944
Cofiniteness of local cohomology modules in the class of modules in dimension less than a fixed integer
Volume 62, no. 1 (2021), pp. 191–198

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 $\operatorname{Hom}_{R}\big(\frac{R}{\mathfrak{a}}, X\big)$ and $\operatorname{Ext}_{R}^{1}\big(\frac{R}{\mathfrak{a}}, X\big)$ 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 $\{\mathfrak{p}\in \operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X)) : \dim\big(\frac{R}{\mathfrak{p}}\big)\geq n\}$ is a finite set for all $i$ when $\operatorname{Ext}_{R}^{i}\big(\frac{R}{\mathfrak{a}}, X\big)$ 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.