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We study the generalized local homology for linearly compact modules which is a generalization of the local homology theory. By duality, we get some properties of the generalized local cohomology and extend well-known properties of the local cohomology theory of Grothendieck.

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Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.