Rigid objects in higher cluster categories

Abstract

We study maximal m-rigid objects in the m-cluster category 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 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.