On modules associated to coalgebra Galois extensions
Abstract
Description
For a given entwining structure $(A,C)_ψ$ involving an algebra $A$, a coalgebra $C$, and an entwining map $ψ: C\otimes A\to A\otimes C$, a category $\M_A^C(ψ)$ of right $(A,C)_ψ$-modules is defined and its structure analysed. In particular, the notion of a measuring of $(A,C)_ψ$ to $(\tA,\tC)_\tpsi$ is introduced, and certain functors between $\M_A^C(ψ)$ 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.
31 pages, LaTeX, uses amscd and amssymb. Some changes in Section 3. Version to appear in J. Algebra
31 pages, LaTeX, uses amscd and amssymb. Some changes in Section 3. Version to appear in J. Algebra