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 Cm(A) of A is the m-left part Lm(A (m)) of the m-replicated algebra of A. Moreover, we obtain a one-toone correspondence between the tilting objects in Cm(A) (that is, the m-clusters) and those tilting modules in modA(m) for which all non projective-injective direct summands lie in Lm(A (m)). 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 AuslanderReiten quiver. Email addresses: [email protected] (I. Assem), [email protected] (T. Brüstle), [email protected] (R. Schiffler), todorov.neu.edu (G. Todorov). 1 Partially supported by the NSERC of Canada 2 Partially supported by the NSERC of Canada and the universities of Sherbrooke Preprint submitted to Elsevier Science 2 February 2008