IngentaConnect On Modules Associated to Coalgebra Galois Extensions

Author: Montgomery T.S.

Source: Journal of Algebra, Volume 215, Number 1, May 1999 , pp. 290-317(28)

Abstract:

For a given entwining structure (A, C) involving an algebra A, a coalgebra C, and an entwining map psi : C otimes A rarr A otimes C, a category M^C_A( psi ) of right (A, C)-modules is defined and its structure analysed. In particular, the notion of a measuring of (A, C) to (Ã, C~)_~ is introduced, and certain functors between M^C_A( psi ) and M^C~_Ã( psi ~) 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 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 rarr B or rarr 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 otimes C as left B-modules and right C-comodules. The relationship between the modules E and is studied in the case when V is finite-dimensional and in the case when the canonical entwining map is bijective. Copyright 1999 Academic Press.

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