Abelian Category of Cominimax and Weakly Cofinite Modules

Abstract

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an arbitrary $R$-module. Let $\mathcal{S}$ be a Serre subcategory of the category of $R$-modules. It is shown that the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $i \geq 0$, if and only if the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $0 \leq i \leq \operatorname{ara}(I)$. As an immediate consequence, we prove that if $R$ is a Noetherian (resp. $(R, \mathfrak{m})$ is a Noetherian local) ring of dimension $d$, then the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $i \geq 0$ if and only if the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $0 \leq i \leq d+1$ (resp. for all $0 \leq i \leq d$). Also it is shown that if $I$ is a principal ideal up to radical, then the category of $I$-cominimax (resp. $I$-weakly cofinite) modules is an Abelian full subcategory of the category of $R$-modules.