The Stacks project

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

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

Then we obtain the surjection

as desired. $\square$