The Stacks project

Lemma 58.73.11. 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 finite locally constant sheaves on $X_{\acute{e}tale}$ is the colimit of the categories of finite locally constant sheaves on $(X_ i)_{\acute{e}tale}$.

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

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

Proof. By Lemma 58.73.10 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 58.51.2 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. $\square$


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