The Stacks project

Lemma 59.73.10. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes with affine transition morphisms. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$.

  1. The category of constructible sheaves of sets on $X_{\acute{e}tale}$ is the colimit of the categories of constructible sheaves of sets on $(X_ i)_{\acute{e}tale}$.

  2. The category of constructible abelian sheaves on $X_{\acute{e}tale}$ is the colimit of the categories of constructible abelian sheaves on $(X_ i)_{\acute{e}tale}$.

  3. Let $\Lambda $ be a Noetherian ring. The category of constructible sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$ is the colimit of the categories of constructible sheaves of $\Lambda $-modules on $(X_ i)_{\acute{e}tale}$.

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

\[ f_0^{-1}\mathcal{G}(U) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} f_{i0}^{-1}\mathcal{G}(U_ i) \]

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 $n$. $\square$

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