# Cartier divisor

On a scheme $X$, a Cartier divisor is a global section of the sheaf ${\mathrm{\pi \x9d\x92\xa6}}^{*}/{\mathrm{\pi \x9d\x92\u037a}}^{*}$, where ${\mathrm{\pi \x9d\x92\xa6}}^{*}$ is the multiplicative sheaf of meromorphic functions, and ${\mathrm{\pi \x9d\x92\u037a}}^{*}$ the multiplicative sheaf of invertible regular functions^{} (the units of the structure sheaf).

More explicitly, a Cartier divisor is a choice of open cover ${U}_{i}$ of $X$, and meromorphic functions ${f}_{i}\beta \x88\x88{\mathrm{\pi \x9d\x92\xa6}}^{*}\beta \x81\u2019({U}_{i})$, such that ${f}_{i}/{f}_{j}\beta \x88\x88{\mathrm{\pi \x9d\x92\u037a}}^{*}\beta \x81\u2019({U}_{i}\beta \x88\copyright {U}_{j})$, along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if ${f}_{i}$ is replaced by $g\beta \x81\u2019{f}_{i}$ with $g\beta \x88\x88{\mathrm{\pi \x9d\x92\u037a}}_{*}$.

Intuitively, the only carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a Weil divisor, and vice versa. On βniceβ (for example, nonsingular over an algebraically closed field) schemes, it does.

Title | Cartier divisor |
---|---|

Canonical name | CartierDivisor |

Date of creation | 2013-03-22 13:52:29 |

Last modified on | 2013-03-22 13:52:29 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 14A99 |