  In previous joint work with Eli Aljadeff we attached a generic Hopf Galois
extension A(H,c) to each twisted algebra H(c) obtained from a Hopf algebra H by
twisting its product with the help of a cocycle c. The algebra A(H,c) is a flat
deformation of H(c) over a "big" central subalgebra B(H,c) and can be viewed as
the noncommutative analogue of a versal torsor in the sense of Serre. After
surveying the results on A(H,c) obtained with Aljadeff, we establish three new
results: we present a systematic method to construct elements of the
commutative algebra B(H,c), we show that a certain important integrality
condition is satisfied by all finite-dimensional Hopf algebras generated by
grouplike and skew-primitive elements, and we compute B(H,c) in the case where
H is the Hopf algebra of a cyclic group.
