On Galois Comodules

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P _A is finitely generated and projective and the evaluation map μ𝒞:Hom ^𝒞 (P, 𝒞) ?  _S P → 𝒞 is an isomorphism (of corings) where S = End ^𝒞 (P). It has been observed that for such comodules the functors ? ?  _A 𝒞 and Hom _A (P, ?) ?  _S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define 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. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.