m-cluster categories and m-replicated algebras

Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category image of A is the m-left part image of the m-replicated algebra of A. Moreover, we obtain a one-to-one correspondence between the tilting objects in image (that is, the m-clusters) and those tilting modules in image for which all non-projective–injective direct summands lie in image. Furthermore, we study the module category of A(m) and show that a basic exceptional module with the correct number of non-isomorphic indecomposable summands is actually a tilting module. We also show how to determine the projective dimension of an indecomposable A(m)-module from its position in the Auslander–Reiten quiver.