Stratifying triangulated categories

Abstract

A notion of stratification is introduced for any compactly generated triangulated category ⊤ endowed with an action of a graded-commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences that follow if ⊤ is stratified by R. Among them are a classification of the localizing subcategories of ⊤ 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 ⊤; and results on the support of the graded R-module of morphisms Hom*⊤(C, D) leading to analogs of the tensor product theorem for support varieties of modular representation of groups.