Melkersson condition on Serre subcategories

Reza Sazeedeh ¹   ; Rasul Rasuli ²

¹ Department of Mathematics Urmia University P.O. Box 165, Urmia, Iran and School of Mathematics Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5746, Tehran, Iran
² Mathematics Department Faculty of Science Payame Noor University (PNU) Tehran, Iran

Colloquium Mathematicum, Tome 144 (2016) no. 2, pp. 289-300

Résumé

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.

Détail

DOI : 10.4064/cm6384-9-2015

Keywords: commutative noetherian ring mathfrak ideal nbsp mathcal subcategory category r modules condition mathfrak defined r modules introduced aghapournahr melkersson order study local cohomology modules relative mathfrak belong mathcal paper define study class mathcal mathfrak consisting modules satisfying nbsp mathfrak mathfrak mathfrak ideals get necessary sufficient condition mathcal satisfy mathfrak mathfrak simultaneously sufficient conditions under which mathcal satisfies mathfrak application investigate local cohomology modules lie serre subcategory