Cartier divisor

Let $X$ be a scheme and let $K_X$ be its sheaf of meromorphic functions, with the injective structure map ${\mathcal{O}}_X\to K_X$ (Sheaf total quotient rings, A sheaf on a topological space). The presheaf $U⟼K_X(U{)}^{\times }\mathrm{/}{\mathcal{O}}_X(U{)}^{\times }$ of abelian groups (units of the two sheaves of rings, the second embedded in the first through the injective structure map) has a sheafification in the sense of Sheafification of a presheaf; the resulting sheaf of abelian groups is denoted $K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times }\mathrm{.}$

A Cartier divisor on $X$ is a global section of $K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times }$. The group of Cartier divisors is denoted $\mathrm{CaDiv}(X)$; its law is induced by the group law of the quotient sheaf, so that the sum of two Cartier divisors is represented by the product of their local meromorphic equations, the zero element $0$ is the class of the constant equation $1$, and the inverse of a divisor is represented by the inverted local equations.

Concretely, a Cartier divisor can be described as follows. Let $\mathcal{U}=\{U_i{\}}_{i\in I}$ be an open cover of $X$ and let $f_i\in K_X(U_i{)}^{\times }$ be a meromorphic unit on $U_i$ for each $i$, with $f_i\mathrm{/}{f}_j\in {\mathcal{O}}_X(U_i\cap U_j{)}^{\times }\text{for all }i,j\mathrm{.}$ The images of the $f_i$ in $(K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times })(U_i)$ agree on the overlaps $U_i\cap U_j$ (their quotient becomes $1$ in the quotient group), so the sheaf axiom glues them to a global section, and a different choice of cover or of representatives $f_i$ (that is, passing to a refinement and multiplying $f_i$ by units of ${\mathcal{O}}_X$ over the pieces) yields the same class. Conversely every global section of the quotient sheaf is locally represented in this way, as the local-equation description of Cartier divisors on this page records.

The principal Cartier divisors are the Cartier divisors that are the images of a global meromorphic unit under the canonical map $\mathrm{\Gamma }(X,K_X^{\times })\to \mathrm{\Gamma }(X,K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times })$. They form a subgroup of $\mathrm{CaDiv}(X)$; a Cartier divisor is principal exactly when it admits a representation by a single global equation on $X$. The sign convention used on this page is that a Cartier divisor with local equation $f$ records a zero of $f$ with positive coefficient and a pole with negative coefficient; the convention is fixed once and for all in the definition of the principal Cartier divisor of a meromorphic unit.

If $X=\mathrm{\varnothing }$ then ${\mathcal{O}}_X=K_X=0$, the quotient sheaf is the zero sheaf and $\mathrm{CaDiv}(\mathrm{\varnothing })=0$; if $X$ is the spectrum of a field, then $K_X={\mathcal{O}}_X=\kappa$ and again $\mathrm{CaDiv}(X)=0$, consistently with the fact that a Cartier divisor measures the failure of a meromorphic unit to be a global unit.