Lemma 31.2.5 (05AF)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.2: Associated points
Lemma 31.2.5 (cite)

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Lemma 31.2.5. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $\text{Ass}(\mathcal{F}) \cap U$ is finite for every quasi-compact open $U \subset X$.

Proof. This is true because the set of associated primes of a finite module over a Noetherian ring is finite, see Algebra, Lemma 10.63.5. To translate from schemes to algebra use that $U$ is a finite union of affine opens, each of these opens is the spectrum of a Noetherian ring (Properties, Lemma 28.5.2), $\mathcal{F}$ corresponds to a finite module over this ring (Cohomology of Schemes, Lemma 30.9.1), and finally use Lemma 31.2.2. $\square$

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