Equivalences between categories of modules and categories of comodules

TL;DR

This paper explores the deep connections between different Galois theories for comodules and establishes equivalences between categories of comodules and modules over firm rings, linking these concepts within a unified framework.

Contribution

It demonstrates the equivalence of various Galois theories for comodules and relates these to category equivalences involving modules over firm rings.

Findings

Different Galois theories for comodules are closely connected.

Categories of comodules over a coring are equivalent to modules over a firm ring.

These equivalences are related to Galois theory for comodules.

Abstract

We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. G\'omez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and [R. Wisbauer, On Galois comodules (2004), to appear in Comm. Algebra.]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.