In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category. The KapustinLi formula describes a CalabiYau structure on the Z 2-graded homotopy category M F (f) of MF (f). No direct combinatorial proof is known as yet. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category.
The compactification is the closure of an embedding (depending on q) of the. We prove a similar characterization for higher cluster categories. We particularly consider the case of the 2-CalabiYau category of the A2 quiver.
We prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles.
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