Lemma 59.71.5 (095G)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.71: Constructible sheaves
Lemma 59.71.5 (cite)

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Lemma 59.71.5. Let $f : X \to Y$ be a morphism of schemes. If $\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\Lambda$ -modules (with $\Lambda$ Noetherian) on $Y_{\acute{e}tale}$ , the same is true for $f^{-1}\mathcal{F}$ on $X_{\acute{e}tale}$ .

Proof. By Lemma 59.71.4 this reduces to the case where $X$ and $Y$ are affine. By Lemma 59.71.2 it suffices to find a finite partition of $X$ by constructible locally closed subschemes such that $f^{-1}\mathcal{F}$ is finite locally constant on each of them. To find it we just pull back the partition of $Y$ adapted to $\mathcal{F}$ and use Lemma 59.64.2. $\square$

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