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...

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba...

For self-orthogonal modules T , the quotient triangulated categories D(A)/K(addT ) are studied, in particular, their relations with the stable categories of some Frobenius categories are investigated. New descriptions of the singularity categories of Gorenstein algebras are obtained; and the derived categories of Gorenstein algebras are explicitly given, inside the stable categories of the grad...

Contents 1. Introduction 1 2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 13 5. Localization via Brown representatbility 23 6. Well generated triangulated categories 31 7. Localization for well generated categories 38 8. Epilogue: Beyond well-generatedness 46 Appendix A. The abelianization of a triangulated category ...

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are ...

The results in this note are all from [Nee01] or [BN93]. For necessary background on cardinals (particularly regular and singular cardinals) see our notes on Basic Set Theory (BST). In our BST notes there is no mention of grothendieck universes, whereas all our notes on category theory (including this one) are implicitly working inside a fixed grothendieck universe U using the conglomerate conv...

We show that a triangulated motivic category admits categorical Thom isomorphisms for vector bundles with an additional structure if and only the generalized cohomology theory represented by tensor unit object classes. also stable $\mathbb{A}^1$-derived does not admit oriented and, more generally, symplectic bundles. In order to do so we compute first homology sheaves of sphere spectrum class i...

A k-linear triangulated category A is called locally finite provided ∑ X∈indA dimk HomA(X,Y ) < ∞ for any indecomposable object Y in A. It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form L̂n: · · · · ̂ n n− 1 2 1 By using this, we prove that the Aus...