Representation dimensions of triangular matrix algebras

Let A be a finite dimensional hereditary algebra over an algebraically closed field be the triangular matrix algebra. We prove that is at most three if A is Dynkin type and is at most four if A is not Dynkin type.Let be the duplicated algebra of A. Let T be a tilting A-module and be a tilting A(1)-module. We show that is representation finite if and only if the full subcategory of is of finite type, where τ is the Auslander–Reiten translation and F(TA) is the torsion-free class of associated with T.Moreover, we also prove that is at most three if A is Dynkin type.