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Principal homogeneous spaces for arbitrary Hopf algebras

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Abstract

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗ B A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

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  1. Mathematisches Institut, Universität München, Theresienstr. 39, D-8000, München 2, FRG

    Hans-Jürgen Schneider

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  1. Hans-Jürgen Schneider
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Schneider, HJ. Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72, 167–195 (1990). https://doi.org/10.1007/BF02764619

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