Modules cofinite and weakly cofinite with respect to an ideal

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal [Formula: see text] of a Noetherian ring [Formula: see text]. It is shown that an [Formula: see text]-module [Formula: see text] is cofinite with respect to [Formula: see text], if and only if, [Formula: see text] is finitely generated for all [Formula: see text], whenever [Formula: see text]. In addition, we show that if [Formula: see text] is finitely generated and [Formula: see text] are weakly Laskerian for all [Formula: see text], then [Formula: see text] are [Formula: see text]-cofinite for all [Formula: see text] and for any minimax submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are finitely generated, where [Formula: see text] is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first [Formula: see text]-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all [Formula: see text]-weakly cofinite modules over [Formula: see text] forms a full Abelian subcategory of the category of modules.

Serre subcategories and the cofiniteness of generalized local cohomology modules with respect to a pair of ideals

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.