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arXiv · 1005.2422

On the Cluster Category of a Marked Surface

Thomas Brüstle·Jie Zhang
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Abstract

We study in this paper the cluster category C(S,M) of a marked surface (S,M). We explicitly describe the objects in C(S,M) as direct sums of homotopy classes of curves in (S,M) and one-parameter families related to closed curves in (S,M). Moreover, we describe the Auslander-Reiten structure of the category C(S,M) in geometric terms and show that the objects without self-extensions in C(S,M) correspond to curves in (S,M) without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

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Thomas Brüstle, Jie Zhang. 2010-06-17. On the Cluster Category of a Marked Surface. https://doi.org/10.2140/ant.2011.5.529

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