111 research outputs found
Acyclic Calabi-Yau categories
We show that an algebraic 2-Calabi-Yau triangulated category over an
algebraically closed field is a cluster category if it contains a cluster
tilting subcategory whose quiver has no oriented cycles. We prove a similar
characterization for higher cluster categories. As a first application, we show
that the stable category of maximal Cohen-Macaulay modules over a certain
isolated singularity of dimension three is a cluster category. As a second
application, we prove the non-acyclicity of the quivers of endomorphism
algebras of cluster-tilting objects in the stable categories of
representation-infinite preprojective algebras. 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.Comment: Introduction rewritten, references updated. 16 page
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