Rigid objects in higher cluster categories

Abstract

We study maximal $m$-rigid objects in the $m$-cluster category ${\mathcal{C}}_H^m$ associated with a finite dimensional hereditary algebra $H$ with $n$ nonisomorphic simple modules. We show that all maximal $m$-rigid objects in these categories have exactly $n$ nonisomorphic indecomposable summands, and that any almost complete $m$-rigid object in ${\mathcal{C}}_H^m$ has exactly $m+1$ nonisomorphic complements. We also show that the maximal $m$-rigid objects and the $m$-cluster tilting objects in these categories coincide, and that the class of finite dimensional algebras associated with maximal $m$-rigid objects is closed under certain factor algebras.

2nd version 17 pages. More details have been added and some proofs have been improved. Some references have also been added

References in corpus (1)

• On triangulated orbit categories