Melkersson condition for extension of Serre subcategories

TL;DR

This paper investigates the Melkersson condition for extending Serre subcategories over noetherian rings, generalizing previous results and exploring properties of local cohomology modules of weakly Laskerian modules.

Contribution

It extends and generalizes existing results on the Melkersson condition, particularly in the context of weakly Laskerian modules and their local cohomology.

Findings

Conditions for local cohomology modules to lie in Serre subcategories

Extension of results on the Melkersson condition

Cofiniteness properties of local cohomology modules

Abstract

Let $R$ be a commutative noetherian ring and let $\frak a$ be an ideal of $R$. In this paper, we study a certain condition, namely $C_{\frak a}$, introduced by Aghapournahr and Melkersson, on the extension of two subcategories of $R$-modules. We extend and generalize some of the main results of Yoshizawa [Y1,Y2]. As an example of extension of subcategories, we study the weakly Laskerian modules and we find some conditions under which the local cohomology modules of a weakly Laskerian module lie in an arbitrary Serre subcategory. Eventually, we investigate the cofiniteness of the local cohomology modules of weakly Laskerian modules.