Lemma 31.25.4. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. In each of the following cases $\mathcal{L}$ has a regular meromorphic section:
$X$ is integral,
$X$ is reduced and any quasi-compact open has a finite number of irreducible components,
$X$ is locally Noetherian and has no embedded points.
Proof. In case (1) let $\eta \in X$ be the generic point. We have seen in Lemma 31.25.3 that $\mathcal{K}_ X$, resp. $\mathcal{K}_ X(\mathcal{L})$ is the constant sheaf with value $\kappa (\eta )$, resp. $\mathcal{L}_\eta $. Since $\dim _{\kappa (\eta )} \mathcal{L}_\eta = 1$ we can pick a nonzero element $s \in \mathcal{L}_\eta $. Clearly $s$ is a regular meromorphic section of $\mathcal{L}$. In case (2) pick $s_\eta \in \mathcal{L}_\eta $ nonzero for all generic points $\eta $ of $X$; this is possible as $\mathcal{L}_\eta $ is a $1$-dimensional vector space over $\kappa (\eta )$. It follows immediately from the description of $\mathcal{K}_ X$ and $\mathcal{K}_ X(\mathcal{L})$ in Lemma 31.25.1 that $s = \prod s_\eta $ is a regular meromorphic section of $\mathcal{L}$. Case (3) is Lemma 31.24.4. $\square$
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