A local duality principle in derived categories of commutative Noetherian rings

TL;DR

This paper generalizes local duality and vanishing theorems for derived categories of Noetherian rings by introducing colocalization functors supported on arbitrary subsets of the spectrum.

Contribution

It introduces colocalization functors with supports in arbitrary subsets, extending classical local duality to a broader context.

Findings

Local duality theorem holds for colocalization functors

Vanishing theorem of Grothendieck type applies to these functors

Generalization to arbitrary subsets of Spec R

Abstract

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors with supports in arbitrary subsets of Spec R, which is a natural generalization of right derived functors of section functors with supports in specialization-closed subsets. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for colocalization functors.