# separated scheme

A scheme $X$ is defined to be a separated scheme if the morphism

$$d:X\to X{\times}_{\mathrm{Spec}\mathbb{Z}}X$$ |

into the fibre product $X{\times}_{\mathrm{Spec}\mathbb{Z}}X$ which is induced by the identity maps^{} $i:X\u27f6X$ in each coordinate is a closed immersion.

Note the similarity^{} to the definition of a Hausdorff topological space. In the situation of topological spaces^{}, a space $X$ is Hausdorff if and only if the diagonal morphism $X\u27f6X\times X$ is a closed embedding^{} of topological spaces. The definition of a separated scheme is very similar^{}, except that the topological product is replaced with the scheme fibre product.

More generally, if $X$ is a scheme over a base scheme $Y$, the scheme $X$ is defined to be *separated* over $Y$ if the diagonal embedding

$$d:X\to X{\times}_{Y}X$$ |

is a closed immersion.

Title | separated scheme |
---|---|

Canonical name | SeparatedScheme |

Date of creation | 2013-03-22 12:50:25 |

Last modified on | 2013-03-22 12:50:25 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 14A15 |

Defines | separated |