A local-global principle for small triangulated categories

TL;DR

This paper develops a local-global principle and stratification framework for small triangulated categories using local cohomology functors, extending concepts from compactly generated categories.

Contribution

It introduces local cohomology functors for cohomological functors on small triangulated categories with ring actions, establishing a new local-global principle and stratification approach.

Findings

Established a local-global principle for small triangulated categories.

Developed a notion of stratification analogous to that in compactly generated categories.

Constructed local cohomology functors for cohomological functors.

Abstract

Local cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category T equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for T and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.