mm-cluster categories and mm-replicated algebras

Let AA be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the mm-cluster category Cm(A)Cm(A) of AA is the mm-left part Lm(A(m))Lm(A(m)) of the mm-replicated algebra of AA. Moreover, we obtain a one-to-one correspondence between the tilting objects in Cm(A)Cm(A) (that is, the mm-clusters) and those tilting modules in modA(m) for which all non-projective–injective direct summands lie in Lm(A(m))Lm(A(m)).Furthermore, we study the module category of A(m)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)A(m)-module from its position in the Auslander–Reiten quiver.