Rational Modules for Corings

Abstract

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gómez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R 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 the author's dissertation measuring α-pairings and can be considered as the corner stone in his 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 results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.

Key Words:

• Corings
• Rational modules
• Comodules
• Entwining structures
• Entwined modules
• Doi-Koppinen structures
• Doi- Koppinen modules

Acknowledgments

Most of the results in this paper are generalizations of results in my dissertation at the Heinrich-Heine Universität (Düsseldorf – Deutschland). I am so grateful to my supervisor Prof. Robert Wisbauer for the continuous support and encouragement. I also thank Tomasz Brzeziński for drawing my attention to the theory of corings and entwined modules during my visit to him in Swansea and for example (1.1). Many thanks go to José Gómez-Torrecillas for his inspiring ideas and for the useful preprints on the subject he sent me.