Lemma 72.7.2 (0EPS)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.7: The Weil divisor class associated to an invertible module
Lemma 72.7.2 (cite)

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Lemma 72.7.2. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$ . Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Let $s \in \mathcal{K}_ X(\mathcal{L})$ be a regular (i.e., nonzero) meromorphic section of $\mathcal{L}$ . Then the sets

$\{ Z \subset X \mid Z \text{ a prime divisor with generic point }\xi \text{ and }s\text{ not in }\mathcal{L}_\xi \}$

and

$\{ Z \subset X \mid Z \text{ is a prime divisor and } \text{ord}_{Z, \mathcal{L}}(s) \not= 0\}$

are locally finite in $X$ .

Proof. There exists a nonempty open subspace $U \subset X$ such that $s$ corresponds to a section of $\Gamma (U, \mathcal{L})$ which generates $\mathcal{L}$ over $U$ . Hence the prime divisors which can occur in the sets of the lemma all correspond to irreducible components of $|X| \setminus |U|$ . Hence Lemma 72.6.1. gives the desired result. $\square$

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