On the derived category of a graded commutative noetherian ring

TL;DR

This paper establishes a correspondence between certain subcategories of the derived category of a graded commutative noetherian ring and subsets of its homogeneous spectrum, with applications to weighted projective schemes.

Contribution

It introduces a bijection linking twist-closed localizing subcategories of the derived category to subsets of the homogeneous spectrum for a broad class of graded rings.

Findings

Bijection between localizing subcategories and homogeneous spectrum subsets

Inclusion-preserving correspondence for graded rings

Application to weighted projective schemes

Abstract

For any graded commutative noetherian ring, where the grading group is abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed localizing subcategories of the derived category, and, on the other hand, subsets of the homogeneous spectrum of prime ideals of the ring. We provide an application to weighted projective schemes.