Galois comodules

Abstract

Galois comodules of a coring are studied. The conditions for a simple comodule to be a Galois comodule are found. A special class of Galois comodules termed principal comodules is introduced. These are defined as Galois comodules that are projective over their comodule endomorphism rings. A complete description of principal comodules in the case a background ring is a field is found. In particular it is shown that a (finitely generated and projective) right comodule of an $A$-coring $\mathcal C$ is principal provided a lifting of the canonical map is a split epimorphism in the category of left $\mathcal C$-comodules. This description is then used to characterise principal extensions or non-commutative principal bundles. Specifically, it is proven that, over a field, any entwining structure consisting of an algebra $A$, a coseparable coalgebra $C$ and a bijective entwining map $\psi$ together with a group-like element in $C$ give rise to a principal extension provided the lifted canonical map is surjective. Induction of Galois and principal comodules via morphisms of corings is described. A connection between the relative injectivity of a Galois comodule and the properties of the extension of endomorphism rings associated to this comodule is revealed.