  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.
