The Stacks project

Lemma 71.10.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying (a), (b), and (c) of Lemma 71.10.8. Then every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ has a regular meromorphic section.

Proof. With notation as in Lemma 71.10.8 the stalk $\mathcal{L}_\eta $ of $\mathcal{L}$ at is defined for all $\eta \in X^0$ and it is a rank $1$ free $\mathcal{O}_{X, \eta }$-module. Pick a generator $s_\eta \in \mathcal{L}_\eta $ for all $\eta \in X^0$. It follows immediately from the description of $\mathcal{K}_ X$ and $\mathcal{K}_ X(\mathcal{L})$ in Lemma 71.10.8 that $s = \prod s_\eta $ is a regular meromorphic section of $\mathcal{L}$. $\square$

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