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Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition $C_{}$, defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class ${}_{}$ consisting of all modules satisfying $C_{}$. If and are ideals of R, we get a necessary and sufficient condition for to satisfy $C_{}$ and $C_{}$ simultaneously. We also find some sufficient...

Let denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either is principal, or R is complete local and is a prime ideal with dim R/ = 1, then the generalized local cohomology module $H_{}^i(M,N)$ is -cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].