Cartier divisor

On a scheme X, a Cartier divisor is a global section of the sheaf 𝒦*/π’ͺ*, where 𝒦* is the multiplicative sheaf of meromorphic functions, and π’ͺ* the multiplicative sheaf of invertible regular functionsMathworldPlanetmath (the units of the structure sheaf).

More explicitly, a Cartier divisor is a choice of open cover Ui of X, and meromorphic functions fiβˆˆπ’¦*⁒(Ui), such that fi/fj∈π’ͺ*⁒(Ui∩Uj), 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 fi is replaced by g⁒fi with g∈π’ͺ*.

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 β€œnice” (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