Coserreness with respect to specialization closed subsets and some Serre subcategories.

Let be a specialization closed subset of Spec R and be a Serre subcategory of Mod R. As a generalization of the notion of cofiniteness, we introduce the concept of -coserreness with respect to (see Definition 4.1). First, as a main result, for some special Serre subcategories , we show that an R -module M with dim R M ≤ 1 is -coserre with respect to if and only if Ext R 0 , 1 (R / , M) ∈ for all ideals ∈ F (). 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 -coserre R -modules M with dim R M ≤ 1 is a full Abelian subcategory of Mod R. Also, for every homologically bounded R -complex X whose homology modules belong to we show that the local cohomology modules H i (X) for all i , are -coserre in all the cases where dim R ≤ 1 , dim R X ≤ 2 − sup X and cd (, X) ≤ 1 − sup X. [ABSTRACT FROM AUTHOR]