Duality Theorems for Graded Algebras and Coalgebras

The concepts of smash product and smash coproduct have been proved to be extremely useful tools in the theory of graded rings and graded coalgebras, and more recently in the theory of action and coactions of a Hopf algebra on an algebra or on a coalgebra. Once these concepts were Ž defined, a natural question appeared: how can the smash product resp. . Ž the smash coproduct be related to a matrix algebra resp. to a matrix . w x coalgebra ? The answer is given by the duality theorems proved in 3 for w x graded rings, and in 5 for graded coalgebras. Various duality theorems Ž w x w x. for algebras have been proved during the last ten years cf. 2 or 10 . Ž The purpose of this paper is to study duality for the smash product resp. . Ž the smash coproduct associated to a G-set and a G-graded ring resp. a . G-graded coalgebra . The mentioned smash product and smash coproduct w x w x w x have been constructed in 10 and 6 . It is shown by an example in 12 that for a G-graded ring R and a left G-set A, we do not have a duality