The Galois Theory of Matrix C-rings |
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Authors: | Tomasz Brzeziński Ryan B Turner |
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Institution: | (1) Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK;(2) Vrije Universiteit Brussel, VUB, Pleinlaan 2, B-1050 Brussel, Belgium |
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Abstract: | A theory of monoids in the category of bicomodules of a coalgebra C or C-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting
of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix
C
-ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite
injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider’s Theorem II Schneider, Isr. J. Math., 72:167–195, 1990], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed.
Dedicated to Stef Caenepeel on the occasion of his 50th birthday. |
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Keywords: | C-ring matrix ring context Galois module bicomodule coalgebra |
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