Stratifying triangulated categories.

A notion of stratification is introduced for any compactly generated triangulated category ⊤ endowed with an action of a graded‐commutative noetherian ring <italic>R</italic>. The utility of this notion is demonstrated by establishing diverse consequences that follow if ⊤ is stratified by <italic>R</italic>. Among them are a classification of the localizing subcategories of ⊤ in terms of subsets of the set of prime ideals in <italic>R</italic>; a classification of the thick subcategories of the subcategory of compact objects in ⊤; and results on the support of the graded <italic>R</italic>‐module of morphisms Hom*⊤(<italic>C</italic>, <italic>D</italic>) leading to analogs of the tensor product theorem for support varieties of modular representation of groups. [ABSTRACT FROM AUTHOR]