Cofiniteness with respect to extension of Serre subcategories

Abstract

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$, $\mathcal{S}$ a Serre subcategory of $R$-modules satisfying the condition $C_{\mathfrak{a}}$ and $\mathcal{N}$ the subcategory of finitely generated $R$-modules. In this paper, we continue the study of $\mathcal{N}\mathcal{S}$-$\mathfrak{a}$-cofinite modules with respect to the extension subcategory $\mathcal{N}\mathcal{S}$, show that some classical results of $\mathfrak{a}$-cofiniteness hold for $\mathcal{N}\mathcal{S}$-$\mathfrak{a}$-cofiniteness in the cases $\dim R=d$ or $\dim R/\mathfrak{a}=d-1$, where $d$ is a positive integer. We also study $\mathcal{N}\mathcal{S}$-$\mathfrak{a}$-cofiniteness of local cohomology modules and the modules ${\mathrm{Ext}}_R^i(N,M)$ and ${\mathrm{Tor}}_i^R(N,M)$.

14 pages, comments welcome