ar X iv : m at h / 02 10 08 9 v 3 [ m at h . Q A ] 4 A pr 2 00 5 Rational Modules for Corings ∗

The so called dense pairings were studied mainly by D. Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with J. Gómez-Torrecillas and J. 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 Y. Doi , M. Koppinen and C. Menini et al. Introduction Let (H,A,C) be a right-right Doi-Koppinen structure over a commutative ring R, M(H)A the corresponding category of Doi-Koppinen modules and A#C the Koppinen opposite smash product. If RC is flat, thenM(H) C A is a Grothendieck category with enough injective objects. A sufficient, however not necessary, condition for M(H)A to embed as a full subcategory of MA#opC∗ is the projectivity of RC [24, Proposition 3.1]. A similar result for a left-right Doi-Koppinen structure (H,A,C) was obtained by Y. Doi [14, 3.1], where the corresponding category of Doi-Koppinen modules AM(H) C was shown to be naturally isomorphic to the category of #-rational #(C,A)-modules. In this paper we show that these results can be obtained under a weaker condition, that RC is locally projective, as corollaries from the more general theory of rational modules for corings over a (not necessarily commutative) ring. Moreover, we show that these categories are of type σ[M ], the theory of which is well developed (e.g. [39]). This extends our results in [3] and [2] on MSC (2000): 16W30