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The Stacks project

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}).

  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 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).

  3. The zero scheme Z(s) is a locally principal closed subspace of X.

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

Proof. Omitted. \square


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