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The Stacks project

[Exposee IX, Proposition 2.14, SGA4]

Lemma 59.74.4. Let $X$ be a quasi-compact and quasi-separated scheme.

  1. Let $\mathcal{F}$ be a constructible sheaf of sets on $X_{\acute{e}tale}$. There exist an injective map of sheaves

    \[ \mathcal{F} \longrightarrow \prod \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{E_ i} \]

    where $f_ i : Y_ i \to X$ is a finite and finitely presented morphism and $E_ i$ is a finite set.

  2. Let $\mathcal{F}$ be a constructible abelian sheaf on $X_{\acute{e}tale}$. There exist an injective map of abelian sheaves

    \[ \mathcal{F} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i} \]

    where $f_ i : Y_ i \to X$ is a finite and finitely presented morphism and $M_ i$ is a finite abelian group.

  3. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$. There exist an injective map of sheaves of modules

    \[ \mathcal{F} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i} \]

    where $f_ i : Y_ i \to X$ is a finite and finitely presented morphism and $M_ i$ is a finite $\Lambda $-module.

Proof. We will reduce this lemma to the Noetherian case by absolute Noetherian approximation. Namely, by Limits, Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits _{t \in T} X_ t$ with each $X_ t$ of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and with affine transition morphisms. By Lemma 59.73.10 the category of constructible sheaves (of sets, abelian groups, or $\Lambda $-modules) on $X_{\acute{e}tale}$ is the colimit of the corresponding categories for $X_ t$. Thus our constructible sheaf $\mathcal{F}$ is the pullback of a similar constructible sheaf $\mathcal{F}_ t$ over $X_ t$ for some $t$. Then we apply the Noetherian case (Lemma 59.74.3) to find an injection

\[ \mathcal{F}_ t \longrightarrow \prod \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{E_ i} \quad \text{or}\quad \mathcal{F}_ t \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i} \]

over $X_ t$ for some finite morphisms $f_ i : Y_ i \to X_ t$. Since $X_ t$ is Noetherian the morphisms $f_ i$ are of finite presentation. Since pullback is exact and since formation of $f_{i, *}$ commutes with base change (Lemma 59.55.3), we conclude. $\square$

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