# Cartier divisor

On a scheme $X$, a Cartier divisor is a global section of the sheaf $\mathcal{K}^{*}/\mathcal{O}^{*}$, where $\mathcal{K}^{*}$ is the multiplicative sheaf of meromorphic functions, and $\mathcal{O}^{*}$ 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 $f_{i}\in\mathcal{K}^{*}(U_{i})$, such that $f_{i}/f_{j}\in\mathcal{O}^{*}(U_{i}\cap U_{j})$, 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 $gf_{i}$ with $g\in\mathcal{O}_{*}$.

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 “nice” (for example, nonsingular over an algebraically closed field) schemes, it does.

Title Cartier divisor CartierDivisor 2013-03-22 13:52:29 2013-03-22 13:52:29 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 14A99