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Israel Journal of Mathematics

, Volume 72, Issue 1–2, pp 167–195 | Cite as

Principal homogeneous spaces for arbitrary Hopf algebras

  • Hans-Jürgen Schneider
Article

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.

Keywords

Hopf Algebra Hopf Subalgebra Hopf Module Comodule Structure Principal Homogeneous Space 
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Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • Hans-Jürgen Schneider
    • 1
  1. 1.Mathematisches InstitutUniversität MünchenMünchen 2FRG

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