A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γW with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then γW coincides with the right derived functor RΓW of the section functor ΓW with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γW with W being an arbitrary subset.