On Modules Associated to Coalgebra Galois Extensions

For a given entwining structure (A, C)ψinvolving an algebraA, a coalgebraC, and an entwining map ψ:C A → A C, a categoryMCA(ψ) of right (A, C)ψ-modules is defined and its structure analysed. In particular, the notion of a measuring of (A, C)ψto (Ã, ) is introduced, and certain functors betweenMCA(ψ) andM Ã( ) induced by such a measuring are defined. It is shown that these functors are inverse equivalences iff they are exact (or one of them faithfully exact) and the measuring satisfies a certain Galois-type condition. Next, left modulesEand right modules associated to aC-Galois extensionAofBare defined. These can be thought of as objects dual to fibre bundles with coalgebraCin the place of a structure group, and a fibreV. Cross-sections of such associated modules are defined as module mapsE → Bor → B. It is shown that they can be identified with suitably equivariant maps from the fibre toA. Also, it is shown that aC-Galois extension is cleft if and only ifA = B Cas leftB-modules and rightC-comodules. The relationship between the modulesEand is studied in the case whenVis finite-dimensional and in the case when the canonical entwining map is bijective