The Stacks project

Lemma 71.7.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. If $D \subset X$ is an effective Cartier divisor, then the canonical section $1_ D$ of $\mathcal{O}_ X(D)$ is regular.

  2. Conversely, if $s$ is a regular section of the invertible sheaf $\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism $\mathcal{O}_ X(D) \to \mathcal{L}$ which maps $1_ D$ to $s$.

The constructions $D \mapsto (\mathcal{O}_ X(D), 1_ D)$ and $(\mathcal{L}, s) \mapsto Z(s)$ give mutually inverse maps

\[ \left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible} \\ \mathcal{O}_ X\text{-module and a regular global section} \end{matrix} \right\} \]

Proof. Omitted. $\square$

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