Melkersson condition on Serre subcategories
Volume 144 / 2016
Abstract
Let $R$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $R$, and let $\mathcal {S}$ be a subcategory of the category of $R$-modules. The condition $C_{\mathfrak a}$, defined for $R$-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to $\mathfrak a$ belong to $\mathcal {S}$. In this paper, we define and study the class $\mathcal {S}_{\mathfrak a}$ consisting of all modules satisfying $C_{\mathfrak a}$. If $\mathfrak a$ and $\mathfrak b$ are ideals of $R$, we get a necessary and sufficient condition for $\mathcal {S}$ to satisfy $C_{\mathfrak a}$ and $C_{\mathfrak b}$ simultaneously. We also find some sufficient conditions under which $\mathcal {S}$ satisfies $C_{\mathfrak a}$. As an application, we investigate when local cohomology modules lie in a Serre subcategory.
