Coinduction functor and simple comodules

TL;DR

This paper introduces a coinduction functor for corings with exact rational functor, establishing a bijective correspondence between simple comodules and simple modules over an endomorphism ring, with applications to group-graded modules.

Contribution

It constructs a new coinduction functor for corings and demonstrates its role in classifying simple comodules via endomorphism rings, extending previous theories.

Findings

Bijective correspondence between simple comodules and modules over endomorphism rings

Construction of a coinduction functor as right adjoint to a hom-functor

Application to group-graded modules

Abstract

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (\emph{coinduction functor}) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of simple right modules over the endomorphism ring. A detailed application to a group-graded modules is also given.