Cofiniteness with respect to extension of Serre subcategories

TL;DR

This paper generalizes the concept of ofiniteness of modules over a noetherian ring by extending it to Serre subcategories satisfying certain conditions, and demonstrates that classical results remain valid in this broader context.

Contribution

It introduces ofiniteness with respect to extension of Serre subcategories, expanding the classical notion and verifying that existing results hold in this generalized setting.

Findings

Classical ofiniteness results hold for the generalized notion in lower dimensions.

Defines and studies ofiniteness relative to extension subcategories.

Generalizes the classical concept of ofiniteness for modules over noetherian rings.

Abstract

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\mathcal{S}$ be an arbitrary Serre subcategory of $R$-modules satisfying the condition $C_{\frak a}$ and let $\mathcal{N}$ be the subcategory of finitely generated $R$-modules. In this paper, we define and study $\mathcal{NS}$-$\frak a$-cofinite modules with respect to the extension subcategory $\mathcal{NS}$ as an generalization of the classical notion, namely $\frak a$-cofinite modules. For the lower dimensions, we show that the classical results of $\frak a$-cofiniteness hold for the new notion.