In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of Toën in [13] for a triangulated category with (left) homological-finite condition.

4.1 Historical Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 4.3 Waldhausen...

We characterize Neeman’s well generated triangulated categories and discuss some of its basic properties. 2000 Mathematics Subject Classification: 18E30 (55P42, 55U35).

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could...

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 ||

By using the approach to Hall algebras arising in homologically finite triangulated categories in [9], we find an “almost” associative multiplication structure for indecomposable objects in a 2-period triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in [6] which provides a way of constructing all symmetrizable Kac-Moody Lie algebras from two periodic...