Lemma 31.24.4 (0EMI)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.24: Meromorphic functions and sections; Noetherian case
Lemma 31.24.4 (cite)

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Lemma 31.24.4. Let $X$ be a locally Noetherian scheme having no embedded points. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Then $\mathcal{L}$ has a regular meromorphic section.

Proof. For each generic point $\eta$ of $X$ pick a generator $s_\eta$ of the free rank $1$ module $\mathcal{L}_\eta$ over the artinian local ring $\mathcal{O}_{X, \eta }$ . It follows immediately from the description of $\mathcal{K}_ X$ and $\mathcal{K}_ X(\mathcal{L})$ in Lemma 31.24.3 that $s = \prod s_\eta$ is a regular meromorphic section of $\mathcal{L}$ . $\square$

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