### Abstract

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ 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 ⊆ 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.

Original language | English |
---|---|

Pages (from-to) | 225-269 |

Number of pages | 45 |

Journal | Journal of Algebra |

Volume | 318 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2007 |

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### Keywords

- Galois extensions
- Hopf algebroids
- Relative injective comodule algebras
- Relative separable functors

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*318*(1), 225-269. https://doi.org/10.1016/j.jalgebra.2007.05.017

**A Schneider type theorem for Hopf algebroids.** / Ardizzoni, A.; Böhm, G.; Menini, C.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 318, no. 1, pp. 225-269. https://doi.org/10.1016/j.jalgebra.2007.05.017

}

TY - JOUR

T1 - A Schneider type theorem for Hopf algebroids

AU - Ardizzoni, A.

AU - Böhm, G.

AU - Menini, C.

PY - 2007/12/1

Y1 - 2007/12/1

N2 - Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ 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 ⊆ 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.

AB - Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ 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 ⊆ 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.

KW - Galois extensions

KW - Hopf algebroids

KW - Relative injective comodule algebras

KW - Relative separable functors

UR - http://www.scopus.com/inward/record.url?scp=35349001173&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35349001173&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2007.05.017

DO - 10.1016/j.jalgebra.2007.05.017

M3 - Article

AN - SCOPUS:35349001173

VL - 318

SP - 225

EP - 269

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -