The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09Y9. Beware of the difference between the letter 'O' and the digit '0'.