The Stacks project

Lemma 59.64.5. Let $X$ be a scheme.

  1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of sets on $X_{\acute{e}tale}$. If $\mathcal{F}$ is finite locally constant, there exists an étale covering $\{ U_ i \to X\} $ such that $\varphi |_{U_ i}$ is the map of constant sheaves associated to a map of sets.

  2. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of abelian groups on $X_{\acute{e}tale}$. If $\mathcal{F}$ is finite locally constant, there exists an étale covering $\{ U_ i \to X\} $ such that $\varphi |_{U_ i}$ is the map of constant abelian sheaves associated to a map of abelian groups.

  3. Let $\Lambda $ be a ring. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$. If $\mathcal{F}$ is of finite type, then there exists an étale covering $\{ U_ i \to X\} $ such that $\varphi |_{U_ i}$ is the map of constant sheaves of $\Lambda $-modules associated to a map of $\Lambda $-modules.

Proof. This holds on any site, see Modules on Sites, Lemma 18.43.3. $\square$

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