Four problems regarding representable functors

TL;DR

This paper explores the properties of representable functors between categories of comodules over corings, establishing equivalences and conditions for functors to be representable, equivalences, separable, or Frobenius, generalizing classical Morita theory.

Contribution

It characterizes when induction functors between comodule categories are representable, equivalences, separable, or Frobenius, unifying and extending Morita theory results.

Findings

Equivalence of the category of representable functors with the opposite of a comodule category.

Necessary and sufficient conditions for induction functors to be representable or equivalences.

Generalization of Morita theorems to categories of comodules over corings.

Abstract

Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\mathcal M}$ the category of left $C$-comodules. The category ${\bf Rep}\, ( {}_{R}^C{\mathcal M}, {}_{S}{\mathcal M} )$ of all representable functors ${}_{R}^C{\mathcal M} \to {}_{S}{\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\otimes_R - : {}_{R}^C\mathcal{M} \to {}_{S}\mathcal{M}$ 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.