The Stacks project

Proof. Proof of (1). Because we have the ascending chain condition for subsheaves of $\mathcal{F}$ (Lemma 59.74.2), it suffices to show that for every point $x \in X$ we can find a map $\varphi : \mathcal{F} \to f_*\underline{E}$ where $f : Y \to X$ is finite and $E$ is a finite set such that $\varphi _{\overline{x}} : \mathcal{F}_{\overline{x}} \to (f_*S)_{\overline{x}}$ is injective. (This argument can be avoided by picking a partition of $X$ as in Lemma 59.71.2 and constructing a $Y_ i \to X$ for each irreducible component of each part.) Let $Z \subset X$ be the induced reduced scheme structure (Schemes, Definition 26.12.5) on $\overline{\{ x\} }$. Since $\mathcal{F}$ is constructible, there is a finite separable extension $K/\kappa (x)$ such that $\mathcal{F}|_{\mathop{\mathrm{Spec}}(K)}$ is the constant sheaf with value $E$ for some finite set $E$. Let $Y \to Z$ be the normalization of $Z$ in $\mathop{\mathrm{Spec}}(K)$. By Morphisms, Lemma 29.53.13 we see that $Y$ is a normal integral scheme. As $K/\kappa (x)$ is a finite extension, it is clear that $K$ is the function field of $Y$. Denote $g : \mathop{\mathrm{Spec}}(K) \to Y$ the inclusion. The map $\mathcal{F}|_{\mathop{\mathrm{Spec}}(K)} \to \underline{E}$ is adjoint to a map $\mathcal{F}|_ Y \to g_*\underline{E} = \underline{E}$ (Lemma 59.73.13). This in turn is adjoint to a map $\varphi : \mathcal{F} \to f_*\underline{E}$. Observe that the stalk of $\varphi$ at a geometric point $\overline{x}$ is injective: we may take a lift $\overline{y} \in Y$ of $\overline{x}$ and the commutative diagram

proves the injectivity. We are not yet done, however, as the morphism $f : Y \to Z$ is integral but in general not finite¹.

To fix the problem stated in the last sentence of the previous paragraph, we write $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ with $Y_ i$ irreducible, integral, and finite over $Z$. Namely, apply Properties, Lemma 28.22.13 to $f_*\mathcal{O}_ Y$ viewed as a sheaf of $\mathcal{O}_ Z$-algebras and apply the functor $\underline{\mathop{\mathrm{Spec}}}_ Z$. Then $f_*\underline{E} = \mathop{\mathrm{colim}}\nolimits f_{i, *}\underline{E}$ by Lemma 59.51.7. By Lemma 59.73.8 the map $\mathcal{F} \to f_*\underline{E}$ factors through $f_{i, *}\underline{E}$ for some $i$. Since $Y_ i \to Z$ is a finite morphism of integral schemes and since the function field extension induced by this morphism is finite separable, we see that the morphism is finite étale over a nonempty open of $Z$ (use Algebra, Lemma 10.140.9; details omitted). This finishes the proof of (1).

The proofs of (2) and (3) are identical to the proof of (1). $\square$