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A Schneider type theorem for Hopf algebroids

Alessandro Ardizzoni·Gabriella Böhm·Claudia Menini
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Abstract

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the non-commutative base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B\subseteq A and also to the equivalence of the category of relative Hopf modules to the category of B-modules. This extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf algebra. Our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor, a notion introduced and analysed hereby. In the first version of this submission, we heavily used the statement that two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule categories. This statement was based on \cite[Brz3,Theorem 2.6], whose proof turned out to contain an unjustified step. In the revised version we return to an earlier definition of a comodule of a Hopf algebroid, that distinguishes between comodules of the two constituent bialgebroids, and modify the statements and proofs in the paper accordingly.

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Alessandro Ardizzoni, Gabriella Böhm, Claudia Menini. 2008-11-01. A Schneider type theorem for Hopf algebroids. https://doi.org/10.1016/j.jalgebra.2007.05.017%3B%2010.1016%2Fj.jalgebra.2008.11.040

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