Melkersson condition on Serre subcategories

Abstract

Let $R$ be a commutative noetherian ring, let $\mathfrak{a}$ and $\mathfrak{b}$ 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 $C_{\mathfrak{a}}$ and $C_{\mathfrak{b}}$ conditions. As an conclusion we show that over a artinian local ring, every Serre subcategory satisfies $C_{\mathfrak{a}}$ 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 $C_{\mathfrak{a}}$ condition. We prove that $C_{\mathfrak{a}}$ condition can be transferred via rings homomorphism. As some applications, we give several results concerning with Serre subcategories in local cohomology theory.