The Image of the Derived Category in the Cluster Category

Abstract

Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for nonhereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In the more general setup of algebras of global dimension at most 2, cluster categories are defined to be these triangulated hulls of the orbit categories. In this paper, we study the image of the natural functor from the bounded, derived category to the cluster category, that is, we investigate how far the orbit category is from being the cluster category. We show that the cluster combinatorics can be worked with in the orbit category, that is, that it is not necessary to consider the entire cluster category. On the other hand, we show that for wide classes of nonpiecewise hereditary algebras the orbit category is never equal to the cluster category.