Lemma 59.74.2 (09YV)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.74: Constructible sheaves on Noetherian schemes
Lemma 59.74.2 (cite)

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Lemma 59.74.2. Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring. Consider inclusions

$\mathcal{F}_1 \subset \mathcal{F}_2 \subset \mathcal{F}_3 \subset \ldots \subset \mathcal{F}$

in the category of sheaves of sets, abelian groups, or $\Lambda$ -modules. If $\mathcal{F}$ is constructible, then for some $n$ we have $\mathcal{F}_ n = \mathcal{F}_{n + 1} = \mathcal{F}_{n + 2} = \ldots$ .

Proof. By Proposition 59.74.1 we see that $\mathcal{F}_ i$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ are constructible. Then the lemma follows from Lemma 59.71.8. $\square$

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