Stratifying triangulated categories

TL;DR

This paper introduces a stratification framework for compactly generated triangulated categories with ring actions, enabling classification of subcategories and support varieties, thus advancing the understanding of their structure and representation theory.

Contribution

It develops a new notion of stratification for triangulated categories with ring actions and demonstrates its utility in classifying subcategories and support varieties.

Findings

Classified localizing subcategories via prime ideals in R

Classified thick subcategories of compact objects

Established an analogue of the tensor product theorem for support varieties

Abstract

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in T; and results concerning the support of the R-module of homomorphisms Hom_T^*(C,D) leading to an analogue of the tensor product theorem for support varieties of modular representation of groups.