# group scheme

A *group scheme* is a group object in the category^{} of schemes. Similarly, if $S$ is a scheme, a *group scheme over $S$* is a group object in the category of schemes over $S$.

As usual with schemes, the points of a group scheme are not the whole story. For example, a group scheme may have only one point over its field of definition and yet not be trivial. The points of the underlying topological space^{} do not form a group under the obvious choice for a group law.

We can view a group scheme $G$ as a “group machine^{}”: given a ring $R$, the set of $R$-points of $G$ forms a group. If $S$ is a scheme that is not affine, we can nevertheless interpret $G$ as a family of groups fibred over $S$.

Title | group scheme |

Canonical name | GroupScheme |

Date of creation | 2013-03-22 14:11:13 |

Last modified on | 2013-03-22 14:11:13 |

Owner | archibal (4430) |

Last modified by | archibal (4430) |

Numerical id | 4 |

Author | archibal (4430) |

Entry type | Definition |

Classification | msc 14K99 |

Classification | msc 14A15 |

Classification | msc 14L10 |

Classification | msc 20G15 |

Related topic | Group |

Related topic | GroupVariety |

Related topic | Category |

Related topic | GroupObject |

Related topic | GroupSchemeOfMultiplicativeUnits |

Related topic | VarietyOfGroups |