Cofiniteness with respect to extension of Serre subcategories at small dimensions

TL;DR

This paper investigates the properties of cofiniteness for modules over noetherian rings concerning Serre subcategories and ideal extensions, focusing on cases where the quotient ring has small dimension.

Contribution

It introduces a new perspective on cofiniteness relative to extension subcategories and a novel dimension concept for modules over rings with small dimension.

Findings

Characterization of $rak a$-cofinite modules for $ ext{dim} R/rak a extless 3

Analysis of cofiniteness with respect to extension of Serre subcategories

Introduction of a new dimension related to cofiniteness

Abstract

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\cS$ be an arbitrary Serre subcategory of $R$-modules and let $\cN$ be the subcategory of finitely generated $R$-modules. In this paper, we study $\cN\cS$-$\frak a$-cofinite modules with respect to the extension subcategory $\cN\cS$ when $\dim R/\frak a\leq 2$. We also study $\frak a$-cofiniteness with respect to a new dimension.