Melkersson condition for extension of Serre subcategories

Abstract

Let R be a commutative noetherian ring and let a be an ideal of R. In this paper, we study a certain condition, namely Ca, 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.