Coserreness with respect to specialization closed subsets and some serre subcategories

Abstract

Let [Formula: see text] be a specialization closed subset of [Formula: see text] and [Formula: see text] be a Serre subcategory of Mod R. As a generalization of the notion of cofiniteness, we introduce the concept of [Formula: see text]-coserreness with respect to [Formula: see text] (see Definition 4.1). First, as a main result, for some special Serre subcategories [Formula: see text], we show that an [Formula: see text]-module [Formula: see text] with [Formula: see text] is [Formula: see text]-coserre with respect to [Formula: see text] if and only if [Formula: see text] for all ideals [Formula: see text]. Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of [Formula: see text]-coserre [Formula: see text]-modules [Formula: see text] with [Formula: see text] is a full Abelian subcategory of Mod R. Also, for every homologically bounded [Formula: see text]-complex [Formula: see text] whose homology modules belong to [Formula: see text] we show that the local cohomology modules [Formula: see text] for all [Formula: see text], are [Formula: see text]-coserre in all the cases where [Formula: see text], [Formula: see text] and [Formula: see text].