Melkersson condition for extension of Serre subcategories

Abstract

Let R be a commutative noetherian ring and let $$\mathfrak {a}$$ be an ideal of R. In this paper, we study a certain condition, namely $$C_{\mathfrak {a}}$$ , introduced by Aghapournahr and Melkersson, on the extension of two subcategories of R-modules. We develop some of the main results of Yoshizawa (Proc Am Math Soc 140(7):2293–2305, 2012; J Commut Algebra 13:137–155, 2021). 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.