The Image of the Derived Category in the Cluster Category

Claire Amiot (1) , Steffen Oppermann (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Institutt for Matematiske Fag [Trondheim]

Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for non-hereditary algebras orbit categories need not be triangulated, and he introduced the notion of trian-gulated 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, i.e. that it is not necessary to consider the entire cluster category. On the other hand we show that for wide classes of non-piecewise hereditary algebras the orbit category is never equal to the cluster category.

Domaines: Théorie des représentations [math.RT]

Claire Amiot, Steffen Oppermann. The Image of the Derived Category in the Cluster Category. International Mathematics Research Notices, 2012, 2013 (4), pp.733-760. ⟨10.1093/imrn/rns010⟩. ⟨hal-01293781⟩