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 $\mathcal{NS}$-$\mathfrak{a}$-cofinite modules with respect to the extension subcategory $\mathcal{NS}$, show that some classical results of $\mathfrak{a}$-cofiniteness hold for $\mathcal{NS}$-$\mathfrak{a}$-cofiniteness in the cases $\mathrm{dim}R=d$ or $\mathrm{dim}R/\mathfrak{a}=d-1$, where $d$ is a positive integer. We also study $\mathcal{NS}$-$\mathfrak{a}$-cofiniteness of local cohomology modules and the modules $\mathrm{Ext}^i_R(N,M)$ and $\mathrm{Tor}_i^R(N,M)$.