SMASH PRODUCTS AND π-H-COMODULES OF LINEAR MAPS FOR HOPF π-COMODULES

With an aim to explore homological algebra for Hopf -coalgebras, this article investigates the HOM-functor and presents the structure theorem for endomorphism algebras of two- sided pi-((H) under bar, (A) under bar)-Hopf modules, where (H) under bar is a Hopf pi-coalgebra and (A) under bar is a right pi-H-comodule algebra. Moreover, we give a duality theorem involving endomorphism rings of ((H) under bar, A)-Hopf modules, " big" smash product and the smash product, where (H) under bar is a cofree Hopf pi-coalgebra and A is a right (H) under bar -comodule algebra. Finally, under the condition "pi-(H) under bar -Galois extensions," we generalize Theorem 2.5 of Menini and Raianu [8] to the setting of Hopf pi-coalgebras.