A MASCHKE-TYPE THEOREM FOR THE GRADED SMASH COPRODUCT C⋊kG

A MASCHKE-TYPE THEOREM FOR THE GRADED SMASH COPRODUCT C⋊kG

M. Cohen and S. Montgomery showed that a Maschke-type theorem for the smash product, which unlike the corresponding result for group actions, does not require any assumptions about the characterstic of the algebra. Our purpose in this paper is a Maschke-type theorem for the graded smash coproduct C⋊kG: let V be a right C⋊kG-comodule and W a C⋊kG-subcomoduleof V which is a C-direct summand of V. Then W is a C⋊kG-direct summand of V. Also this result is equivalent to the following : let V be a graded right C-comodule and W a graded subcomodule of V which has a complement as a C-subcomodule of V. Then W has a graded complement.