Rational Modules for Corings

The so called dense pairings were studied mainly by D. Radford in his work on coreflexive coalegbras over fileds [Rad73]. They were generalized in [G-T98] and [AG-TL2001] to the so called rational pairings over commutative ground rings to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A −→ C∗. Such pairings, satisfying the so called α-condition, were called in [Abu2001] measuring α-pairings and can be considered as the corner stone in the author’s study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending some of our results in [AG-TL2001] and [Abu2001] to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g. Doi-Koppinen modules, alternative DoiKoppinen modules) generalizing results of Y. Doi [Doi94], M. Koppinen [Kop95] and C. Menini et al. (e.g. [MZ97], [MSTW2001], [MTW2001]).