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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Let  be a abelian ctegory, and (A,B) be a complete hereditary cotorsion pair on . We define relative cohomology groups based on (A,B) in the category of N-complexes on . Especially we are devoted to consider the balancedness of the relative cohomology functors.

Let a be an ideal of a commutative noetherian ring R and M an R-module with Cosupport in V(a). We show that M is a-coartinian if and only if Ext i R (R/a, M) is artinian for all 0 i cd(a, M), which provides a computable finitely many steps to examine a-coartinianness. We also consider the duality of Hartshorne's questions: for which rings R and ideals a are the modules H a i (M) a-coartinian for all i 0; whether the category C(R, a) coa of a-coartinian modules is an Abelian subcategory of the category of all R-modules, and establish affirmative answers to these questions in the case cd(a, R) 1 and dimR/a 1.