Lemma 59.73.5. Let $X$ be a quasi-compact and quasi-separated scheme. The category of constructible sheaves of sets is the full subcategory of $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ consisting of sheaves $\mathcal{F}$ which are coequalizers

$\xymatrix{ \mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_0 \ar[r] & \mathcal{F}}$

such that $\mathcal{F}_ i$, $i = 0, 1$ is a finite coproduct of sheaves of the form $h_ U$ with $U$ a quasi-compact and quasi-separated object of $X_{\acute{e}tale}$.

Proof. In the proof of Lemma 59.73.2 we have seen that sheaves of this form are constructible. For the converse, suppose that for every constructible sheaf of sets $\mathcal{F}$ we can find a surjection $\mathcal{F}_0 \to \mathcal{F}$ with $\mathcal{F}_0$ as in the lemma. Then we find our surjection $\mathcal{F}_1 \to \mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0$ because the latter is constructible by Lemma 59.71.6.

By Topology, Lemma 5.28.7 we may choose a finite stratification $X = \coprod _{i \in I} X_ i$ such that $\mathcal{F}$ is finite locally constant on each stratum. We will prove the result by induction on the cardinality of $I$. Let $i \in I$ be a minimal element in the partial ordering of $I$. Then $X_ i \subset X$ is closed. By induction, there exist finitely many quasi-compact and quasi-separated objects $U_\alpha$ of $(X \setminus X_ i)_{\acute{e}tale}$ and a surjective map $\coprod h_{U_\alpha } \to \mathcal{F}|_{X \setminus X_ i}$. These determine a map

$\coprod h_{U_\alpha } \to \mathcal{F}$

which is surjective after restricting to $X \setminus X_ i$. By Lemma 59.64.4 we see that $\mathcal{F}|_{X_ i} = h_ V$ for some scheme $V$ finite étale over $X_ i$. Let $\overline{v}$ be a geometric point of $V$ lying over $\overline{x} \in X_ i$. We may think of $\overline{v}$ as an element of the stalk $\mathcal{F}_{\overline{x}} = V_{\overline{x}}$. Thus we can find an étale neighbourhood $(U, \overline{u})$ of $\overline{x}$ and a section $s \in \mathcal{F}(U)$ whose stalk at $\overline{x}$ gives $\overline{v}$. Thinking of $s$ as a map $s : h_ U \to \mathcal{F}$, restricting to $X_ i$ we obtain a morphism $s|_{X_ i} : U \times _ X X_ i \to V$ over $X_ i$ which maps $\overline{u}$ to $\overline{v}$. Since $V$ is quasi-compact (finite over the closed subscheme $X_ i$ of the quasi-compact scheme $X$) a finite number $s^{(1)}, \ldots , s^{(m)}$ of these sections of $\mathcal{F}$ over $U^{(1)}, \ldots , U^{(m)}$ will determine a jointly surjective map

$\coprod s^{(j)}|_{X_ i} : \coprod U^{(j)} \times _ X X_ i \longrightarrow V$

Then we obtain the surjection

$\coprod h_{U_\alpha } \amalg \coprod h_{U^{(j)}} \to \mathcal{F}$

as desired. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).