  In previous work, to each Hopf algebra H and each invertible right
two-cocycle on H, Eli Aljadeff and the first-named author attached a subalgebra
B of the free commutative Hopf algebra S generated by the coalgebra underlying
H; the algebra B is the subalgebra of coinvariants of a generic Hopf Galois
extension. In this paper we give conditions under which S is faithfully flat,
or even free, as a B-module. We also show that B is generated as an algebra by
certain elements arising from the theory of polynomial identities for comodule
algebras developped jointly with Aljadeff.
