The Stacks project

Lemma 42.24.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim _\delta (X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$ be a nonzero global section. Then

\[ \text{div}_\mathcal {L}(s) = [Z(s)]_{n - 1} \]

in $Z_{n - 1}(X)$ and

\[ c_1(\mathcal{L}) \cap [X] = [Z(s)]_{n - 1} \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 1}(X)$.

Proof. Let $Z \subset X$ be an integral closed subscheme of $\delta $-dimension $n - 1$. Let $\xi \in Z$ be its generic point. Choose a generator $s_\xi \in \mathcal{L}_\xi $. Write $s = fs_\xi $ for some $f \in \mathcal{O}_{X, \xi }$. By definition of $Z(s)$, see Divisors, Definition 31.14.8 we see that $Z(s)$ is cut out by a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ such that $\mathcal{I}_\xi = (f)$. Hence $\text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{Z(s), \xi }) = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{X, \xi }/(f)) = \text{ord}_{\mathcal{O}_{X, x}}(f)$ as desired. $\square$

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