An approach to Hopf algebras via Frobenius coordinates II

We make a study of finitely generated, projective Hopf algebras over commutative rings from the point of view of P-Frobenius algebras. Defining modular function as the composite of counit with Nakayama automorphism, we establish the Radford formula (Amer. J. Math. 98 (1976) 333–335) [34] by means of comparing and transforming a Frobenius system. We then study when such Hopf algebras are separable and strongly separable, their Hopfsubalgebras as Frobenius extensions of a third kind, their Drinfeld double and a change to a ground ring with trivial Picard group.