Cofiniteness with respect to extension of Serre subcategories

TL;DR

This paper extends the concept of cofiniteness in module theory to subcategories related to Serre subcategories, establishing classical results in new contexts and analyzing local cohomology and Ext/Tor modules.

Contribution

It introduces and studies $ ext{NS}$-$ ext{a}$-cofiniteness for modules relative to extension subcategories, generalizing classical cofiniteness results.

Findings

Classical $ ext{a}$-cofiniteness results hold for $ ext{NS}$-$ ext{a}$-cofiniteness in certain dimension cases.

Established $ ext{NS}$-$ ext{a}$-cofiniteness of local cohomology modules.

Analyzed $ ext{Ext}^i_R(N,M)$ and $ ext{Tor}_i^R(N,M)$ modules in this context.

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 $\mathrm{Tor}_i^R(N,M)$.