Lemma 31.14.9. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf. Let $s \in \Gamma (X, \mathcal{L})$.

1. Consider closed immersions $i : Z \to X$ such that $i^*s \in \Gamma (Z, i^*\mathcal{L})$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set.

2. For any morphism of schemes $f : Y \to X$ we have $f^*s = 0$ in $\Gamma (Y, f^*\mathcal{L})$ if and only if $f$ factors through $Z(s)$.

3. The zero scheme $Z(s)$ is a locally principal closed subscheme.

4. The zero scheme $Z(s)$ is an effective Cartier divisor if and only if $s$ is a regular section of $\mathcal{L}$.

Proof. Omitted. $\square$

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