On cluster-tilting graphs for hereditary categories

Abstract

Let k be an algebraically closed field and H a Hom- and Ext-finite hereditary abelian k-category with tilting objects. It is proved that the cluster-tilting graph associated with H is connected. As a consequence, we establish the connectedness of the tilting graph for the category cohX of coherent sheaves over a weighted projective line X of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel and Unger, which has immediate applications in cluster algebras. For instance, we deduce that there is a bijection between the set of isomorphism classes of indecomposable rigid objects of the cluster category CX of cohX and the set of cluster variables of the cluster algebra AX associated with cohX.