Morita theory of comodules over corings

TL;DR

This paper extends Morita theory to comodules over corings, establishing equivalences between categories of comodules and firm modules for non-unital subrings, broadening the scope beyond finitely generated projective cases.

Contribution

It generalizes Morita equivalence results to non-finitely generated comodules over corings, connecting comodules with firm modules for non-unital subrings, and extends structure theorems.

Findings

Established equivalences between categories of comodules and firm modules.

Extended structure theorems beyond finitely generated projective cases.

Proved a strong structure theorem for firmly projective comodules of coseparable corings.

Abstract

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $\Sigma$ of an $A$-coring $\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\cC$ or the comodule $\Sigma$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of $\cC$-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra $^\cC(\Sigma)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.