Weakly cofiniteness of local cohomology modules

Abstract

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a system of ideals of [Formula: see text] and [Formula: see text]. Let [Formula: see text] be an [Formula: see text]-module (not necessary [Formula: see text]-torsion) such that [Formula: see text], then the [Formula: see text]-module [Formula: see text] is weakly Laskerian, for all [Formula: see text], if and only if the [Formula: see text]-module [Formula: see text] is weakly Laskerian for [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] an [Formula: see text]-module such that [Formula: see text] is weakly Laskerian for all [Formula: see text]. We prove that if the [Formula: see text]-module [Formula: see text] is [Formula: see text] for all [Formula: see text], then [Formula: see text] is [Formula: see text]-weakly cofinite for all [Formula: see text], and for any [Formula: see text] (or minimax) submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are weakly Laskerian. Let [Formula: see text] be a finitely generated [Formula: see text]-module. We also prove that [Formula: see text] and [Formula: see text] are [Formula: see text]-weakly cofinite for all [Formula: see text] and [Formula: see text] whenever [Formula: see text] is weakly Laskerian and [Formula: see text] is [Formula: see text] for all [Formula: see text]. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.