Strong connections and the relative Chern-Galois character for corings

Details Strong connections and the relative Chern-Galois character for corings Strong connections and the relative Chern-Galois character for corings

2005-03-22

arxiv.org/abs/math/0503469v3

Int. Math. Res. Notices, 2005:42 (2005) 2579-2625

Mathematics

Rings and Algebras

34 pages, LaTeX. v3: Minor corrections in Examples 2.5 and 3.11

Scientific paper

The Chern-Galois theory is developed for corings or coalgebras over non-commutative rings. As the first step the notion of an entwined extension as an extension of algebras within a bijective entwining structure over a non-commutative ring is introduced. A strong connection for an entwined extension is defined and it is shown to be closely related to the Galois property and to the equivariant projectivity of the extension. A generalisation of the Doi theorem on total integrals in the framework of entwining structures over a non-commutative ring is obtained, and the bearing of strong connections on properties such as faithful flatness or relative injectivity is revealed. A family of morphisms between the K0-group of the category of finitely generated projective comodules of a coring and even relative cyclic homology groups of the base algebra of an entwined extension with a strong connection is constructed. This is termed a relative Chern-Galois character. Explicit examples include the computation of a Chern-Galois character of depth 2 Frobenius split (or separable) extensions over a separable algebra R. Finitely generated and projective modules are associated to an entwined extension with a strong connection, the explicit form of idempotents is derived, the corresponding (relative) Chern characters are computed, and their connection with the relative Chern-Galois character is explained.

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