Lemma 71.7.7. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. Let s \in \Gamma (X, \mathcal{L}).
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.
For any morphism of algebraic spaces f : Y \to X over S we have f^*s = 0 in \Gamma (Y, f^*\mathcal{L}) if and only if f factors through Z(s).
The zero scheme Z(s) is a locally principal closed subspace of X.
The zero scheme Z(s) is an effective Cartier divisor on X if and only if s is a regular section of \mathcal{L}.
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