Projectivity and freeness over comodule algebras

Let H H be a Hopf algebra and A A an H H -simple right H H -comodule algebra. It is shown that under certain hypotheses every ( H , A ) (H,A) -Hopf module is either projective or free as an A A -module and A A is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for H H -simple H H -module algebras.