On Modules Associated to Coalgebra Galois Extensions

✍ Scribed by Tomasz Brzeziński

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 255 KB
• Volume: 215
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7738

✦ Synopsis

For a given entwining structure A, C involving an algebra A, a coalgebra C, C Ž . Ž . and an entwining map : C m A ª A m C, a category M of right A, C -A modules is defined and its structure analysed. In particular, the notion of a ˜Ž . Ž . measuring of A, C to A, C is introduced, and certain functors between C C Ž . Ž . M and M induced by such a measuring are defined. It is shown that these à A Ž

. 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 ª B or E ª 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 s B m C as left B-modules and right C-comodules. The relationship between the modules E and E is studied in the case when V is finite-dimensional and in the case when the canonical entwining map is bijective.