Cluster combinatorics of d-cluster categories
Abstract
We study the cluster combinatorics of cluster tilting objects in cluster categories. By using mutations of maximal rigid objects in cluster categories which are defined similarly for cluster tilting objects, we prove the equivalences between cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of triangles of cluster tilting objects in [IY], we prove that any almost complete cluster tilting object has exactly complements, compute the extension groups between these complements, and study the middle terms of these triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established in [BMRRT] to cluster categories. They are applied to the Fomin-Reading's generalized cluster complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2], and to that of infinite root systems [Zh3].
Cite
@article{arxiv.0712.1381,
title = {Cluster combinatorics of d-cluster categories},
author = {Yu Zhou and Bin Zhu},
journal= {arXiv preprint arXiv:0712.1381},
year = {2009}
}
Comments
correted many typos according to the referee's comments, final version to appear in J. Algebra