Vol. 5, No. 4, 2011

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On the cluster category of a marked surface without punctures

Thomas Brüstle and Jie Zhang

Vol. 5 (2011), No. 4, 529–566

We study the cluster category C(S,M) of a marked surface (S,M) without punctures. 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 noncontractible 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.

cluster category, marked surface
Mathematical Subject Classification 2000
Primary: 16G99
Secondary: 16G20, 57N05, 57M50, 16G70
Received: 14 May 2010
Revised: 13 August 2010
Accepted: 12 September 2010
Published: 21 December 2011

Proposed: Andrei Zelevinsky
Seconded: Bernd Sturmfels, Victor Reiner
Thomas Brüstle
Département de Mathématiques
Université de Sherbrooke
Sherbrooke  J1K 2R1
Department of Mathematics
Bishop’s University
Sherbrooke  J1M 1Z7
Jie Zhang
Département de Mathématiques
Université de Sherbrooke
Sherbrooke  J1K 2R1