Cluster categories for marked surfaces: punctured case. (English) Zbl 1405.16024
In the paper under review, the authors study the cluster category \(\mathcal C(\mathbf T)\) associated to a marked surface \(\mathbf S\) with punctures and with non-empty boundary. Constructing and using skewed-gentle algebras (see [Ch. Geiss and J. A. de la Peña, Boll. Soc. Mat. Mexicana 5, No. 2, 307–326 (1999; Zbl 0959.16013)]) associated with so-called admissible triangulations \(\mathbf T\) of \(\mathbf S\), they prove that there is a bijection between tagged curves in \(\mathbf S\) and string objects in the category \(\mathcal C(\mathbf T)\). As an application, the authors provide interpretations of dimension of the first extension group in the cluster category \(\mathcal C(\mathbf T)\) as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. They also prove that the exchange graph of cluster tilting objects in \(\mathcal C(\mathbf T)\) is isomorphic to the exchange graph of tagged triangulations of \(\mathbf S\) and hence it is connected.
Reviewer: Piotr Malicki (Toruń)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
Citations:
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