The Stacks project

Proof. By Lemma 59.72.1 there is a partition $\coprod _ i X_ i$ such that $\pi _ i : j^{-1}(X_ i) \to X_ i$ is finite étale. The restriction of $h_ U$ to $X_ i$ is $h_{j^{-1}(X_ i)}$ which is finite locally constant by Lemma 59.64.4. For cases (2) and (3) we note that

by Lemmas 59.70.5 and 59.70.7. Thus it suffices to show the lemma for $\pi : Y \to X$ finite étale. This is Lemma 59.64.3. $\square$

Proof. Let $\mathcal{B}$ be the collection of quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$. By Modules on Sites, Lemma 18.30.7 any sheaf of sets is a filtered colimit of sheaves of the form

with $V_ j$ and $U_ i$ quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$. By Lemmas 59.73.1 and 59.71.6 these coequalizers are constructible. This proves (1).

Let $\Lambda $ be a Noetherian ring. By Modules on Sites, Lemma 18.30.7 $\Lambda $-modules $\mathcal{F}$ is a filtered colimit of modules of the form

with $V_ j$ and $U_ i$ quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$. By Lemmas 59.73.1 and 59.71.6 these cokernels are constructible. This proves (3).

Proof of (2). First write $\mathcal{F} = \bigcup \mathcal{F}[n]$ where $\mathcal{F}[n]$ is the $n$-torsion subsheaf. Then we can view $\mathcal{F}[n]$ as a sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules and apply (3). $\square$

Proof. One implication follows from Lemma 59.71.5. For the converse, assume $f^{-1}\mathcal{F}$ is constructible. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a filtered colimit of constructible sheaves (of sets, abelian groups, or modules) using Lemma 59.73.2. Since $f^{-1}$ is a left adjoint it commutes with colimits (Categories, Lemma 4.24.5) and we see that $f^{-1}\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f^{-1}\mathcal{F}_ i$. By Lemma 59.71.8 we see that $f^{-1}\mathcal{F}_ i \to f^{-1}\mathcal{F}$ is surjective for all $i$ large enough. Since $f$ is surjective we conclude (by looking at stalks using Lemma 59.36.2 and Theorem 59.29.10) that $\mathcal{F}_ i \to \mathcal{F}$ is surjective for all $i$ large enough. Thus $\mathcal{F}$ is the quotient of a constructible sheaf $\mathcal{G}$. Applying the argument once more to $\mathcal{G} \times _\mathcal {F} \mathcal{G}$ or the kernel of $\mathcal{G} \to \mathcal{F}$ we conclude using that $f^{-1}$ is exact and that the category of constructible sheaves (of sets, abelian groups, or modules) is preserved under finite (co)limits or (co)kernels inside $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$, $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$, $\textit{Ab}(Y_{\acute{e}tale})$, $\textit{Ab}(X_{\acute{e}tale})$, $\textit{Mod}(Y_{\acute{e}tale}, \Lambda )$, and $\textit{Mod}(X_{\acute{e}tale}, \Lambda )$, see Lemma 59.71.6. $\square$

Proof. By Lemma 59.71.4 it suffices to check this Zariski locally on $Y$ and by Lemma 59.73.3 we may replace $Y$ by an étale cover (the construction of $f_*$ commutes with étale localization). A finite étale morphism is étale locally isomorphic to a disjoint union of isomorphisms, see Étale Morphisms, Lemma 41.18.3. Thus, in the case of sheaves of sets, the lemma says that if $\mathcal{F}_ i$, $i = 1, \ldots , n$ are constructible sheaves of sets, then $\prod _{i = 1, \ldots , n} \mathcal{F}_ i$ is too. This is clear. Similarly for sheaves of abelian groups and modules. $\square$

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. $\square$

Proof. In the proof of Lemma 59.73.2 we have seen modules of this form are constructible. Since the category of constructible modules is abelian (Lemma 59.71.6) it suffices to prove that given a constructible module $\mathcal{F}$ there is a surjection

for some affine objects $U_ i$ in $X_{\acute{e}tale}$. By Modules on Sites, Lemma 18.30.7 there is a surjection

with $U_ i$ affine and the direct sum over a possibly infinite index set $I$. For every finite subset $I' \subset I$ set

By the very definition of constructible sheaves, the set $T_{I'}$ is a constructible subset of $X$. We want to show that $T_{I'} = \emptyset $ for some $I'$. Since every stalk $\mathcal{F}_{\overline{x}}$ is a finite type $\Lambda $-module and since $\Psi $ is surjective, for every $x \in X$ there is an $I'$ such that $x \not\in T_{I'}$. In other words we have $\emptyset = \bigcap _{I' \subset I\text{ finite}} T_{I'}$. Since $X$ is a spectral space by Properties, Lemma 28.2.4 the constructible topology on $X$ is quasi-compact by Topology, Lemma 5.23.2. Thus $T_{I'} = \emptyset $ for some $I' \subset I$ finite as desired. $\square$

Proof. This follows from Lemma 59.73.6 applied with $\Lambda = \mathbf{Z}/n\mathbf{Z}$ and the fact that, since $X$ is quasi-compact, every constructible abelian sheaf is annihilated by some positive integer $n$ (details omitted). $\square$

Proof. The case of sheaves of sets. By Lemma 59.73.5 it suffices to prove the lemma for $h_ U$ where $U$ is a quasi-compact and quasi-separated object of $X_{\acute{e}tale}$. Recall that $\mathop{\mathrm{Mor}}\nolimits (h_ U, \mathcal{G}) = \mathcal{G}(U)$. Hence the result follows from Sites, Lemma 7.17.7.

In the case of abelian sheaves or sheaves of modules, the result follows in the same way using Lemmas 59.73.7 and 59.73.6. For the case of abelian sheaves, we add that $\mathop{\mathrm{Mor}}\nolimits (j_{U!}\underline{\mathbf{Z}/n\mathbf{Z}}, \mathcal{G})$ is equal to the $n$-torsion elements of $\mathcal{G}(U)$. $\square$

Proof. It suffices to prove this when $X$ and $Y$ are affine by Lemma 59.71.4. By Lemmas 59.55.3 and 59.73.3 we may base change to any affine scheme surjective over $X$. By Lemma 59.72.3 this reduces us to the case of a finite étale morphism (because a thickening leads to an equivalence of étale topoi and even small étale sites, see Theorem 59.45.2). The finite étale case is Lemma 59.73.4. $\square$

Proof. Proof of (1). Denote $f_ i : X \to X_ i$ the projection maps. There are 3 parts to the proof corresponding to “faithful”, “fully faithful”, and “essentially surjective”.

Faithful. Choose $0 \in I$ and let $\mathcal{F}_0$, $\mathcal{G}_0$ be constructible sheaves on $X_0$. Suppose that $a, b : \mathcal{F}_0 \to \mathcal{G}_0$ are maps such that $f_0^{-1}a = f_0^{-1}b$. Let $E \subset X_0$ be the set of points $x \in X_0$ such that $a_{\overline{x}} = b_{\overline{x}}$. By Lemma 59.71.7 the subset $E \subset X_0$ is constructible. By assumption $X \to X_0$ maps into $E$. By Limits, Lemma 32.4.10 we find an $i \geq 0$ such that $X_ i \to X_0$ maps into $E$. Hence $f_{i0}^{-1}a = f_{i0}^{-1}b$.

Fully faithful. Choose $0 \in I$ and let $\mathcal{F}_0$, $\mathcal{G}_0$ be constructible sheaves on $X_0$. Suppose that $a : f_0^{-1}\mathcal{F}_0 \to f_0^{-1}\mathcal{G}_0$ is a map. We claim there is an $i$ and a map $a_ i : f_{i0}^{-1}\mathcal{F}_0 \to f_{i0}^{-1}\mathcal{G}_0$ which pulls back to $a$ on $X$. By Lemma 59.73.5 we can replace $\mathcal{F}_0$ by a finite coproduct of sheaves represented by quasi-compact and quasi-separated objects of $(X_0)_{\acute{e}tale}$. Thus we have to show: If $U_0 \to X_0$ is such an object of $(X_0)_{\acute{e}tale}$, then

where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. This is a special case of Theorem 59.51.3.

Essentially surjective. We have to show every constructible $\mathcal{F}$ on $X$ is isomorphic to $f_ i^{-1}\mathcal{F}$ for some constructible $\mathcal{F}_ i$ on $X_ i$. Applying Lemma 59.73.5 and using the results of the previous two paragraphs, we see that it suffices to prove this for $h_ U$ for some quasi-compact and quasi-separated object $U$ of $X_{\acute{e}tale}$. In this case we have to show that $U$ is the base change of a quasi-compact and quasi-separated scheme étale over $X_ i$ for some $i$. This follows from Limits, Lemmas 32.10.1 and 32.8.10.

Proof of (3). The argument is very similar to the argument for sheaves of sets, but using Lemma 59.73.6 instead of Lemma 59.73.5. Details omitted. Part (2) follows from part (3) because every constructible abelian sheaf over a quasi-compact scheme is a constructible sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules for some $n$. $\square$

Proof. By Lemma 59.73.10 the functor in each case is fully faithful. By the same lemma, all we have to show to finish the proof in case (1) is the following: given a constructible sheaf $\mathcal{F}_ i$ on $X_ i$ whose pullback $\mathcal{F}$ to $X$ is finite locally constant, there exists an $i' \geq i$ such that the pullback $\mathcal{F}_{i'}$ of $\mathcal{F}_ i$ to $X_{i'}$ is finite locally constant. By assumption there exists an étale covering $\mathcal{U} = \{ U_ j \to X\} _{j \in J}$ such that $\mathcal{F}|_{U_ j} \cong \underline{S_ j}$ for some finite set $S_ j$. We may assume $U_ j$ is affine for all $j \in J$. Since $X$ is quasi-compact, we may assume $J$ finite. By Lemma 59.51.2 we can find an $i' \geq i$ and an étale covering $\mathcal{U}_{i'} = \{ U_{i', j} \to X_{i'}\} _{j \in J}$ whose base change to $X$ is $\mathcal{U}$. Then $\mathcal{F}_{i'}|_{U_{i', j}}$ and $\underline{S_ j}$ are constructible sheaves on $(U_{i', j})_{\acute{e}tale}$ whose pullbacks to $U_ j$ are isomorphic. Hence after increasing $i'$ we get that $\mathcal{F}_{i'}|_{U_{i', j}}$ and $\underline{S_ j}$ are isomorphic. Thus $\mathcal{F}_{i'}$ is finite locally constant. The proof in cases (2) and (3) is exactly the same. $\square$

Proof. This is true because for every étale morphism $U \to X$ with $U \not= \emptyset $ the point $\eta $ is in the image. $\square$

Proof. Proof of (1). Let $x \in X$ be a point. Let $\mathcal{O}^{sh}_{X, \overline{x}}$ be a strict henselization of $\mathcal{O}_{X, x}$. By More on Algebra, Lemma 15.45.6 we see that $\mathcal{O}^{sh}_{X, \overline{x}}$ is a normal domain. Hence $\mathop{\mathrm{Spec}}(K) \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}})$ is irreducible. It follows that the stalk $(g_*\underline{E})_{\overline{x}}$ is equal to $E$, see Theorem 59.53.1.

Proof of (2). Since $g$ factors through $j$ there is a map $j_*\underline{E} \to g_*\underline{E}$. This map is injective because for every scheme $V$ étale over $X$ the set $\mathop{\mathrm{Spec}}(K) \times _ X V$ is dense in $U \times _ X V$. On the other hand, we have a map $\underline{E} \to j_*\underline{E}$ and 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. Choose $i \in I$. Since $\mathcal{F}|_\eta $ is zero by assumption, we see that there exists an $i'(i) \geq i$ such that $\mathcal{F}_ i|_\eta \to \mathcal{F}_{i'(i)}|_\eta $ is zero, see Lemma 59.71.8. Then $\mathcal{G}_ i = \mathop{\mathrm{Im}}(\mathcal{F}_ i \to \mathcal{F}_{i'(i)})$ is a constructible abelian sheaf (Lemma 59.71.6) whose stalk at $\eta $ is zero. Hence the support $E_ i$ of $\mathcal{G}_ i$ is a constructible subset of $X$ not containing $\eta $. Since $\eta $ is a generic point of an irreducible component of $X$, we see that $\eta \not\in Z_ i = \overline{E_ i}$ by Topology, Lemma 5.15.15. Define a new directed set $I'$ by using the set $I$ with ordering defined by the rule $i_1$ is bigger or equal to $i_2$ if and only if $i_1 \geq i'(i_2)$. Then the sheaves $\mathcal{G}_ i$ form a system over $I'$ with colimit $\mathcal{F}$ and the proof is complete.

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