  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.
