Cartier divisor

Definition

Let $X$ be a scheme with structure sheaf $\mathcal{O}_X$. For an open set $U \subseteq X$, let $S(U)$ be the set of $s \in \mathcal{O}_X(U)$ whose germ $s_x$ is a non-zero-divisor in $\mathcal{O}_{X,x}$ for every $x \in U$. The sheaf of total quotient rings $\mathcal{K}_X$ is the sheaf associated to the presheaf $U \mapsto S(U)^{-1}\mathcal{O}_X(U)$. For an integral scheme it is the constant sheaf of the function field; in general it is built from the total ring of fractions. Let $\mathcal{K}_X^*$ and $\mathcal{O}_X^*$ be the sheaves of units of $\mathcal{K}_X$ and of $\mathcal{O}_X$. A Cartier divisor on $X$ is a global section of the quotient sheaf $\mathcal{K}_X^* / \mathcal{O}_X^*$.

Equivalently, a Cartier divisor is given by an open cover $(U_i)$ of $X$ and sections $f_i \in \mathcal{K}_X^*(U_i)$ such that $f_i / f_j \in \mathcal{O}_X^*(U_i \cap U_j)$ for all $i, j$. Two such families $(U_i, f_i)$ and $(V_k, g_k)$ define the same divisor when $f_i / g_k \in \mathcal{O}_X^*(U_i \cap V_k)$ for all $i, k$.