The graded centers of derived discrete algebras

Abstract

We describe in the paper the graded centers of the derived categories of the derived discrete algebras. In particular, we prove that if $A$ is a derived discrete algebra, then the reduced part of the graded center of the derived category of $A$ is nontrivial if and only if $A$ has infinite global dimension. Moreover, we show that the nilpotent part of the graded center is controlled by the objects for which the Auslander--Reiten translation coincides with a power of the suspension functor.