The Image of the Derived Category in the Cluster Category

Claire Amiot, Steffen Oppermann
2012 International mathematics research notices  
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 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
more » ... rom 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. This means that to study cluster combinatorics inside the cluster category, it is enough to work in the derived category or the orbit category. These latter categories are more accessible to explicit computations than cluster categories. We then focus on the question of when the orbit category coincides with the cluster category. The most ambitious hope one could have in this direction is the following: Conjecture 1.3. The orbit category coincides with the cluster category for an algebra Λ if and only if Λ is piecewise hereditary. Note that this conjecture, combined with Theorem 1.1, is of a similar flavor to the following question. Question 1.4 (Skowroński [Sko, Question 1]). Let Λ be a finite dimensional algebra, T(Λ) = Λ DΛ the trivial extension of Λ by its k-dual DΛ, which may be considered as a graded algebra by putting Λ
doi:10.1093/imrn/rns010 fatcat:usnvn27cobbs3pw635ompibewm