As Koenig and Zhu showed, quotient of a triangulated category by a maximal 1-orthogonal subcategory becomes an abelian category. In this paper, we generalize this result to a maximal n-orthogonal subcategory for an arbitrary positive integer n.

We prove the relation between the triangulated categories of graded B-branes on simple singularities and the derived categories of representations of Dynkin quivers. The proof is based on the theory of weighted projective lines by Geigle and Lenzing and Orlov’s theorem on the relation between triangulated categories of graded B-branes and the derived category of coherent sheaves.

We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum–Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christe...

The bounded derived category of coherent sheaves D(X) is a natural triangulated category which can be associated with an algebraic variety X. It happens sometimes that two different varieties have equivalent derived categories of coherent sheaves D(X) ≃ D(Y ). There arises a natural question: can one say anything about motives of X and Y in that case? The first such example (see [4]) – abelian ...

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that similar to spaces the derived category obtained by inverting the quasiisomorphisms is natura...

We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category. Moreover, the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures. Along the way, we give a ...

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi-Yau triangulated categories, in particular stable categories of maximal Cohen-Macaul...

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we ...

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial ...