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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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