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