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Galois theory for Hopf algebroids

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Un'estensioneB(A di algebre su un anello commutativok è unaH-estensione per unL-bialgebroideH seA è unaH-comodulo algebra eB è la sottoalgebra dei suoi coinvarianti. Essa èH-Galois se l'applicazione canonicaA ⊗A B→A ⊗L H è un isomorfismo o, equivalentemente, se il coanello canonicoA ⊗L H:A è un coanello di Galois. Nel caso di un algebroide di Hopf\(\mathcal{H} = \left( {\mathcal{H}_L \mathcal{H}_R , S} \right)\) si dimostra che ogniH R-estensione è unaH L-estensione. Se l'antipode è biiettivo allora si dimostra che anche le nozioni di estensioniH R-Galois eH L-Galois coincidono.

I risultati per le strutture biiettive entwining sono estesi alle strutture entwining su algebre non commutative, al fine di dimostrare un teorema simile al Teorema dii Kreimer-Takeuchi per un Hopf algebroideH proiettivo finitamento generato con antipode biiettivo. Il teorema afferma che ogni estensioneH-GaloisB ⊂A è proiettiva e seA èk-piatto allora la suriettività dell'applicazione canonica è sufficiente a garantire la proprietà di Galois.

La teoria di Morita, sviluppata per i coanelli da Caenepeel, Vercruysse e Wang, viene applicata per ottenere criteri equivalenti per la proprietà di Galois per estensioni di algebroidi di Hopf. Questo conduce a risultati analoghi, per algebroidi di Hopf, a quelli ottenuti da Doi per estensioni di algebre di Hopf e da Cohen Fishman e Montgomery nel caso degli algebroidi di Hopf Frobenius.

Abstract

An extensionB⊂A of algebras over a commutative ringk is anH-extension for anL-bialgebroidH ifA is anH-comodule algebra andB is the subalgebra of its coinvariants. It isH-Galois if the canonical mapA ⊗B A →A ⊗L H is an isomorphism or, equivalently, if the canonical coringA ⊗L H:A is a Galois coring.

In the case of Hopf algebroid\(\mathcal{H} = \left( {\mathcal{H}_L \mathcal{H}_R , S} \right)\) anyH R-extension is shown to be also anH L-extension. If the antipode is bijective then also the notions ofH R-Galois extensions and ofH L-Galois extensions are proven to coincide.

Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroidH with bijective antipode. It states that anyH-Galois extensionB⊂A is projective, and ifA isk-flat then already the surjectivity of the canonical map implies the Galois property.

The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

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  1. Gabriella Böhm

    Present address: Research Institute for Particle and Nuclear Physics, Budapest, P.O.B. 49, H-1525, Budapest 114, Hungary

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    Böhm, G. Galois theory for Hopf algebroids. Ann. Univ. Ferrara 51, 233–262 (2005). https://doi.org/10.1007/BF02824833

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