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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. [ABSTRACT FROM PUBLISHER]

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer. [ABSTRACT FROM AUTHOR]

The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form. [ABSTRACT FROM AUTHOR]

We give a short proof of Auslander’s formula relating the covariant and the contravariant defect of a short exact sequence in a module category. An immediate consequence is the classical Auslander–Reiten formula relating Ext and Hom. [Copyright &y& Elsevier]

Abstract: The localising subcategories of the derived category of the cochains on the classifying space of a finite group are classified. They are in one to one correspondence with the subsets of the set of homogeneous prime ideals of the cohomology ring . [Copyright &y& Elsevier]