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Let [Formula: see text] be an ideal of a commutative noetherian ring [Formula: see text] and [Formula: see text] an [Formula: see text]-module with Cosupport in [Formula: see text]. We show that [Formula: see text] is [Formula: see text]-coartinian if and only if [Formula: see text] is artinian for all [Formula: see text], which provides finite steps to examine [Formula: see text]-coartinianess. We also consider the duality of Hartshorne’s questions: for which rings [Formula: see text] and ideals [Formula: see text] are the modules [Formula: see text] [Formula: see text]-coartinian for every artinian [Formula: see text]-module [Formula: see text] and all [Formula: see text]; whether the category [Formula: see text] of [Formula: see text]-coartinian modules is an abelian subcategory of the category of [Formula: see text]-modules, and establish affirmative answers to these questions in the case [Formula: see text] and [Formula: see text].

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For the bounded derived category of an abelian category, bounds of the dimension with respect to a complete hereditary cotorsion pair are given. We also characterize levels of DG-modules and study how levels involved in a recollement of derived categories over DG-rings are related.