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 $\psi: C\otimes A\to A\otimes C$, 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 $(\tA,\tC)_\tpsi$ is introduced, and certain functors between $\M_A^C(\psi)$ and $\M_\tA^\tC(\tpsi)$ 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 $\bar{E}$ 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 $E\to B$ or $\bar{E}\to B$. 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 $A=B\tens C$ as left $B$-modules and right $C$-comodules. The relationship between the modules $E$ and $\bar{E}$ is studied in the case when $V$ is finite-dimensional and in the case when the canonical entwining map is bijective.