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

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.

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.

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.

## Comments (0)