Gr(?)bner-Shirshov Basis Of The Derived Hall Algebra Of Type A_n

We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Gr?bner-Shirshov basis of the Ringel-Hall algebra,and the corresponding irreducible elements forms a PBW type basis.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations among the iso-classes of indecomposable objects in the bounded derived category Db(An)using the Auslander-Reiten quiver of Db(An),and then we prove that all possible compositions among these skew commutator relations are trivial.Finally,as an application,we give a PBW type basis of DH(An).