Melkersson condition on Serre subcategories (Q2804292)

Let \(\mathfrak{a}\) denote an ideal of a commutative Noetherian ring \(R\). A class of \(R\)-modules \(S\) satisfies the condition \(C_{\mathfrak{a}}\) on an \(R\)-module \(M\) if it is of \(\mathfrak{a}\)-torsion and \(0:_M \mathfrak{a} \in S\) implies that \(M \in S\). This notion was introduced by \textit{L. Melkersson} [Math. Proc. Camb. Philos. Soc. 107, No. 2, 267--271 (1990; Zbl 0709.13002)] for \(S\) the class of Artinian \(R\)-modules. Let \(\mathcal{S}\) denote a subcategory of the category of \(R\)-modules. In the paper the authors define and study the class \(\mathcal{S}_{\mathfrak{a}}\) of all modules of \(\mathcal{S}\) satisfying \(C_{\mathfrak{a}}\). For two ideals \(\mathfrak{a}, \mathfrak{b}\) there are necessary and sufficient conditions for \(\mathcal{S}\) to satisfy \(C_{\mathfrak{a}}\) and \(C_{\mathfrak{b}}\) simultaneously. Moreover, there is a sufficient condition under which \(\mathcal{S}\) satisfies \(C_{\mathfrak{a}}\). As an application the authors investigate when local cohomology modules lie in a Serre subcategory.