[Paper Review] Strong connections and the relative Chern-Galois character for corings

Gabriella Böhm, Tomasz Brzeziński|ArXiv.org|Mar 22, 2005

Algebraic structures and combinatorial models19 references4 citations

TL;DR

This paper develops a relative Chern-Galois character for entwined extensions over non-commutative rings using strong connections in corings, generalizing non-commutative geometry tools to depth 2 Frobenius and separable extensions. The key contribution is a family of maps from the Grothendieck group of finitely generated projective comodules to even relative cyclic homology, which becomes independent of the strong connection choice under suitable flatness and splitting conditions.

ABSTRACT

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.

Motivation & Objective

• To extend the Chern-Galois character formalism from fields to non-commutative base rings within entwining structures.
• To establish a link between strong connections, Galois properties, and equivariant projectivity in entwined extensions.
• To construct a relative Chern-Galois character mapping K-theory of comodules to relative cyclic homology.
• To show that under flatness and splitting conditions, the character becomes independent of the choice of strong connection.
• To provide explicit computations of the character for cleft Hopf algebroid extensions and depth 2 Frobenius split extensions.

Proposed method

• Introduces entwined extensions as algebras over a non-commutative ring within a bijective entwining structure.
• Defines a strong T-connection in a bijective entwining structure to generalize the notion of connection in Hopf-Galois theory.
• Uses the generalized Cuntz-Quillen theorem to relate strong connections to equivariant projectivity and the Galois property.
• Constructs even cycles in the T-relative cyclic complex of the base algebra B from finitely generated projective comodules of the coring C.
• Defines the relative Chern-Galois character as a family of maps from K₀(C) to even T-relative cyclic homology groups HC₂ₙ(B|T).
• Applies the canonical epimorphism λ*: HC*(B) → HC*(B|T) and uses the independence of the Chern character on idempotent representatives to show invariance under connection choice.

Experimental results

Research questions

• RQ1 How can the Chern-Galois character be generalized to entwined extensions over non-commutative base rings?
• RQ2 What is the relationship between strong connections, the Galois property, and equivariant projectivity in entwined extensions?
• RQ3 Under what conditions does the relative Chern-Galois character become independent of the choice of strong connection?
• RQ4 How can explicit forms of the Chern-Galois character be computed for cleft Hopf algebroid extensions and depth 2 Frobenius split extensions?
• RQ5 What is the connection between the relative Chern-Galois character and the relative Chern character of the base algebra B?

Key findings

• A strong T-connection exists if and only if the entwined extension is Galois and equivariantly projective, as established in Theorem 3.7.
• The relative Chern-Galois character is constructed as a family of maps from K₀(C) to even T-relative cyclic homology groups HC₂ₙ(B|T), with explicit formulae derived in Theorem 5.4.
• For cleft Hopf algebroid extensions and depth 2 Frobenius split extensions, the character is explicitly computed and shown to coincide with the relative Chern character of B.
• Under the assumptions of splitting as a B-T bimodule, local projectivity of A over T, flatness, and surjectivity of the map (5.1), the relative Chern-Galois character becomes independent of the choice of strong connection, as proven in Theorem 5.14.
• The value of the relative Chern-Galois character equals the relative Chern character of B when the extension is split and satisfies the flatness and projectivity conditions, due to the independence of the Chern character on idempotent representatives.
• The construction of idempotents for finitely generated projective B-modules associated to entwined extensions with strong connections is explicitly derived in Theorem 5.10, enabling the computation of the relative Chern character.