Partial tilting modules over m-replicated algebras

Let A be a hereditary algebra over an algebraically closed field k and A(m) be the m-replicated algebra of A. Given an A(m)-module T, we denote by δ(T) the number of non-isomorphic indecomposable summands of T. In this paper, we prove that a partial tilting A(m)-module T is a tilting A(m)-module if and only if δ(T)=δ(A(m)), and that every partial tilting A(m)-module has complements. As an application, we deduce that the tilting quiver KA(m) of A(m) is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras.