We first introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This unifies the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then define the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those fo...

We study the properties of torsion pairs in triangulated category by introducing the notions of d-Ext-projectivity and d-Ext-injectivity. In terms of -mutation of torsion pairs, we investigate the properties of torsion pairs in triangulated category C D    U Z D Z D

We generalise Yoshino’s definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara’s charac...

Grothendieck first defined the notion of a “motif” as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of...

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories ...

Contents 1. Introduction 1 2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 14 5. Localization via Brown representability 24 6. Well generated triangulated categories 31 7. Localization for well generated categories 39 8. Epilogue: Beyond well generatedness 47 Appendix A. The abelianization of a triangulated category 4...

A recollement describes one triangulated category T as “glued together” from two others, S and U. The definition is not symmetrical in S and U, but this note shows how S and U can be interchanged when T has a Serre functor. A recollement of triangulated categories S, T, U is a diagram of triangulated functors S i∗ // T j∗ // i ||

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differe...

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differe...