The Stacks project

Proof. Proof of (1). Let $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{F}$ a constructible sheaf of sets on $X_{\acute{e}tale}$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta $ such that $\mathcal{G}|_ U$ is locally constant. To do this we may replace $X$ by an étale neighbourhood of $\eta $. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible.

Say $\mathcal{F} = \underline{S}$ for some finite set $S$. Then $S' = \mathcal{G}_{\overline{\eta }} \subset S$ say $S' = \{ s_1, \ldots , s_ t\} $. Pick an étale neighbourhood $(U, \overline{u})$ of $\overline{\eta }$ and sections $\sigma _1, \ldots , \sigma _ t \in \mathcal{G}(U)$ which map to $s_ i$ in $\mathcal{G}_{\overline{\eta }} \subset S$. Since $\sigma _ i$ maps to an element $s_ i \in S' \subset S = \Gamma (X, \mathcal{F})$ we see that the two pullbacks of $\sigma _ i$ to $U \times _ X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma _ i$ comes from a section of $\mathcal{G}$ over the open $\mathop{\mathrm{Im}}(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{S'} \subset \mathcal{G} \subset \underline{S}$. Then we see that $\underline{S'} = \mathcal{G}$ by Lemma 59.73.12.

Let $\mathcal{F} \to \mathcal{Q}$ be a surjection with $\mathcal{F}$ a constructible sheaf of sets on $X_{\acute{e}tale}$. Then set $\mathcal{G} = \mathcal{F} \times _\mathcal {Q} \mathcal{F}$. By the first part of the proof we see that $\mathcal{G}$ is constructible as a subsheaf of $\mathcal{F} \times \mathcal{F}$. This in turn implies that $\mathcal{Q}$ is constructible, see Lemma 59.71.6.

Proof of (3). we already know that constructible sheaves of modules form a weak Serre subcategory, see Lemma 59.71.6. Thus it suffices to show the statement on submodules.

Let $\mathcal{G} \subset \mathcal{F}$ be a submodule of a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta $ such that $\mathcal{G}|_ U$ is locally constant. To do this we may replace $X$ by an étale neighbourhood of $\eta $. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible.

Say $\mathcal{F} = \underline{M}$ for some finite $\Lambda $-module $M$. Then $M' = \mathcal{G}_{\overline{\eta }} \subset M$. Pick finitely many elements $s_1, \ldots , s_ t$ generating $M'$ as a $\Lambda $-module. (This is possible as $\Lambda $ is Noetherian and $M$ is finite.) Pick an étale neighbourhood $(U, \overline{u})$ of $\overline{\eta }$ and sections $\sigma _1, \ldots , \sigma _ t \in \mathcal{G}(U)$ which map to $s_ i$ in $\mathcal{G}_{\overline{\eta }} \subset M$. Since $\sigma _ i$ maps to an element $s_ i \in M' \subset M = \Gamma (X, \mathcal{F})$ we see that the two pullbacks of $\sigma _ i$ to $U \times _ X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma _ i$ comes from a section of $\mathcal{G}$ over the open $\mathop{\mathrm{Im}}(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{M'} \subset \mathcal{G} \subset \underline{M}$. Then we see that $\underline{M'} = \mathcal{G}$ by Lemma 59.73.12.

Proof of (2). This follows in the usual manner from (3). Details omitted. $\square$

Proof. By Proposition 59.74.1 we see that $\mathcal{F}_ i$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ are constructible. Then the lemma follows from Lemma 59.71.8. $\square$

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). $\square$

Proof. We will reduce this lemma to the Noetherian case by absolute Noetherian approximation. Namely, by Limits, Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits _{t \in T} X_ t$ with each $X_ t$ of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and with affine transition morphisms. By Lemma 59.73.10 the category of constructible sheaves (of sets, abelian groups, or $\Lambda $-modules) on $X_{\acute{e}tale}$ is the colimit of the corresponding categories for $X_ t$. Thus our constructible sheaf $\mathcal{F}$ is the pullback of a similar constructible sheaf $\mathcal{F}_ t$ over $X_ t$ for some $t$. Then we apply the Noetherian case (Lemma 59.74.3) to find an injection

over $X_ t$ for some finite morphisms $f_ i : Y_ i \to X_ t$. Since $X_ t$ is Noetherian the morphisms $f_ i$ are of finite presentation. Since pullback is exact and since formation of $f_{i, *}$ commutes with base change (Lemma 59.55.3), we conclude. $\square$

Proof. Proof of (1). We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ with $\mathcal{F}_ i$ constructible abelian by Lemma 59.73.2. By Proposition 59.74.1 the image $\mathcal{F}'_ i \subset \mathcal{F}$ of the map $\mathcal{F}_ i \to \mathcal{F}$ is constructible. Thus $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}'_ i$ and the support of $\mathcal{F}'_ i$ is contained in $E$. Since the support of $\mathcal{F}'_ i$ is constructible (by our definition of constructible sheaves), we see that its closure is also contained in $E$, see for example Topology, Lemma 5.23.6.

The proof in case (2) is exactly the same and we omit it. $\square$