Cartier divisor

On a scheme $X$, a Cartier divisor is a global section of the sheaf ${𝒦}^{*}/{𝒪}^{*}$, where ${𝒦}^{*}$ is the multiplicative sheaf of meromorphic functions, and ${𝒪}^{*}$ 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 $f_i\in {𝒦}^{*}(U_i)$, such that $f_i/f_j\in {𝒪}^{*}(U_i\cap U_j)$, 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 $g{f}_i$ with $g\in {𝒪}_{*}$.

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.