# Weil divisors on schemes

Let $X$ be a noetherian integral separated scheme such that every local ring ${\mathcal{O}}_{x}$ of $X$ of dimension one is regular (such a scheme $X$ is said to be regular in codimension one, or non-singular in codimension one).

###### Definition.

A prime divisor^{} on $X$ is a closed integral subscheme $Y$ of codimension one. We define an abelian group^{} $\mathrm{Div}\mathit{}\mathrm{(}X\mathrm{)}$ generated by the prime divisors on $X$. A Weil divisor is an element of $\mathrm{Div}\mathit{}\mathrm{(}X\mathrm{)}$. Thus, a Weil divisor $\mathrm{W}$ can be written as:

$$\mathcal{W}=\sum {n}_{Y}Y$$ |

where the sum is over all the prime divisors $Y$ of $X$, the ${n}_{Y}$ are integers and only finitely many of them are non-zero. A degree of a divisor^{} is defined to be $\mathrm{deg}\mathit{}\mathrm{(}\mathrm{W}\mathrm{)}\mathrm{=}\mathrm{\sum}{n}_{Y}$. Finally, a divisor is said to be effective if ${n}_{Y}\mathrm{\ge}\mathrm{0}$ for all the prime divisors $Y$.

For more information, see Hartshorne’s book listed in the bibliography for algebraic geometry.

Title | Weil divisors on schemes |
---|---|

Canonical name | WeilDivisorsOnSchemes |

Date of creation | 2013-03-22 15:34:08 |

Last modified on | 2013-03-22 15:34:08 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14C20 |

Related topic | BibliographyForAlgebraicGeometry |

Defines | prime divisor |

Defines | effective divisor |

Defines | regular in codimension one |