Duality and Rational Modules in Hopf Algebras over Commutative Rings,

Let A be an algebra over a commutative ring R. If R is noetherian and A° is pure in RA, then the categories of rational left A-modules and right A°-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner–Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of A° in RA.