Representable Functors for Corings

Abstract

We address four problems regarding representable functors and give answers to them for functors connecting the category of comodules over a coring to the category of modules over a ring. A functor's property of being Frobenius is restated as a particular case of its representability by imposing the predefinition of the object of representability. Let R, S be two rings, C an R-coring and the category of left C-comodules. The category of all representable functors is shown to be equivalent to the opposite of the category . For U an (S, R)-bimodule we give necessary and sufficient conditions for the induction functor to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings.