On modules associated to coalgebra Galois extensions

For a given entwining structure (A, C)(psi) involving an algebra A, a coalgebra C, and an entwining map psi: C x A --> A x C, a category M-A(C)(psi) of right (A, C)(psi)- modules is defined and its structure analysed. In particular, the notion of a measuring of (A, C)(psi) to ((A) over tilde, (C) over tilde)(<(psi)over tilde>) is introduced, and certain functors between M-A(C)(psi) and M-(A) over tilde((C) over tilde)(<(psi)over tilde>) 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 modules E and right modules E associated to a C-Galois extension A of B are defined. These can be thought of as objects dual to fibre bundles with coalgebra C in the place of a structure group, and a fibre V. Cross-sections of such associated modules are defined as module maps E --> B or (E) over bar --> B. It is shown that they can be identified with suitably equivariant maps from the fibre to A. Also, it is shown that a C-Galois extension is cleft if and only if A = B x C as left B-modules and right C-comodules. The relationship between the modules E and (E) over bar is studied in the case when V is finite-dimensional and in the case when the canonical entwining map is bijective. (C) 1999 Academic Press.