A Maschke type theorem for Hopf $\pi$-comodules

Abstract

Let $H$ be a Hopf $\pi$-coalgebra and let $A$ be a right group $H$-comodule algebra with a total integral $\phi$. In this article we will find some sufficient conditions under which an epimorphism of $(H, A)$-Hopf $\pi$-comodule splits if it splits $A$-linearly. As an application, we obtain a characterization for an $(H, A)$-Hopf $\pi$-comodule to be projective as an $A$-module, generalizing the one of the Maschke type theorem found in [D2].