Lemma 31.27.2 (02SG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.27: The Weil divisor class associated to an invertible module
Lemma 31.27.2 (cite)

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Lemma 31.27.2. Let $X$ be a locally Noetherian integral scheme. 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 subscheme $U \subset X$ 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 are all irreducible components of $X \setminus U$. Hence Lemma 31.26.1. gives the desired result. $\square$

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