Lemma 59.74.5 (0F0N)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.74: Constructible sheaves on Noetherian schemes
Lemma 59.74.5 (cite)

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Lemma 59.74.5. Let $X$ be a Noetherian scheme. Let $E \subset X$ be a subset closed under specialization.

Let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$ whose support is contained in $E$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ is a filtered colimit of constructible abelian sheaves $\mathcal{F}_ i$ such that for each $i$ the support of $\mathcal{F}_ i$ is contained in a closed subset contained in $E$.
Let $\Lambda $ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$ whose support is contained in $E$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ is a filtered colimit of constructible sheaves of $\Lambda $-modules $\mathcal{F}_ i$ such that for each $i$ the support of $\mathcal{F}_ i$ is contained in a closed subset contained in $E$.

Proof. Proof of (1). We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ with $\mathcal{F}_ i$ constructible abelian by Lemma 59.73.2. By Proposition 59.74.1 the image $\mathcal{F}'_ i \subset \mathcal{F}$ of the map $\mathcal{F}_ i \to \mathcal{F}$ is constructible. Thus $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}'_ i$ and the support of $\mathcal{F}'_ i$ is contained in $E$. Since the support of $\mathcal{F}'_ i$ is constructible (by our definition of constructible sheaves), we see that its closure is also contained in $E$, see for example Topology, Lemma 5.23.6.

The proof in case (2) is exactly the same and we omit it. $\square$

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