Melkersson condition on Serre subcategories

TL;DR

This paper characterizes when Serre subcategories of modules over a noetherian ring satisfy Melkersson conditions related to ideals, with applications to local cohomology and properties over artinian rings.

Contribution

It provides necessary and sufficient conditions for Serre subcategories to satisfy Melkersson conditions, including closure properties and transferability via ring homomorphisms.

Findings

Over artinian local rings, all Serre subcategories satisfy the $C_{\frak a}$ condition.

The subcategory $\Ss_{\frak a}$ is closed under module extensions.

The $C_{\frak a}$ condition can be transferred through ring homomorphisms.

Abstract

Let $R$ be a commutative noetherian ring, let $\frak a$ and $\frak b$ be two ideals of $R$; and let $\Ss$ be a Serre subcategory of $R$-modules. We give a necessary and sufficient condition by which $\Ss$ satisfies $C_{\frak a}$ and $C_{\frak b}$ conditions. As an conclusion we show that over a artinian local ring, every Serre subcategory satisfies $C_{\frak a}$ condition. We also show that $\Ss_{\frak a}$ is closed under extension of modules. If $\Ss$ is a torsion subcategory, we prove that $S$ satisfies $C_{\frak a}$ condition. We prove that $C_{\frak a}$ condition can be transferred via rings homomorphism. As some applications, we give several results concerning with Serre subcategories in local cohomology theory.