On Galois comodules

Abstract

Generalising the notion of Galois corings, Galois comodules were introduced as comodules $P$ over an $A$-coring $\cC$ for which $P_A$ is finitely generated and projective and the evaluation map $μ_\cC:\Hom^\cC(P,\cC)\ot_SP\to \cC$ is an isomorphism (of corings) where $S=\End^\cC(P)$. It was observed that for such comodules the functors $\Hom_A(P,-)\ot_SP$ and $-\ot_A\cC$ from the category of right $A$-modules to the category of right $\cC$-comodules are isomorphic. In this note we call modules $P$ with this property {\em Galois comodules} without requiring $P_A$ to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. These comodules are close to being generators and have some common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.