Freeness and divisibility for right $H$-simple left $H$-comodule algebras over a pointed Hopf algebra $H$

Let $H$ be a pointed Hopf algebra and let $A$ be a right $H$-simple left $H$-comodule algebra. We show that every relative $(H,A)$-Hopf module is free as an $A$-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative $(H,A)$-Ho