Cofiniteness with Respect to the Class of Modules in Dimension less than a Fixed Integer

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $n$ a non-negative integer, $\mathfrak{a}$ an ideal of $R$ with $\dim(R/\mathfrak{a}) \leq n+1$, and $X$ an arbitrary $R$-module. In this paper, we prove the following results: (i) If $X$ is an $\mathfrak{a}$-torsion $R$-module such that $\operatorname{Hom}_{R}(R/\mathfrak{a},X)$ and $\operatorname{Ext}_{R}^{1}(R/\mathfrak{a},X)$ are $\operatorname{FD}_{\lt n}$ $R$-modules, then $X$ is an $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-module; (ii) The category of $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-modules is an Abelian category; (iii) $\operatorname{H}^{i}_{\mathfrak{a}}(X)$ is an $(\operatorname{FD}_{\lt n},\mathfrak{a})$-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 $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{\lt n}$ $R$-module for all $i$. We observe that, among other things, $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X))$ is a finite set for all $i$ whenever $R$ is a semi-local ring with $\dim(R/\mathfrak{a}) \leq 2$ and $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{\lt 1}$ $R$-module for all $i$.