The Stacks project

Lemma 18.42.3. Let $\mathcal{C}$ be a site with a final object $X$.

  1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of sets on $\mathcal{C}$. If $\mathcal{F}$ is finite locally constant, there exists a 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 $\mathcal{C}$. If $\mathcal{F}$ is finite locally constant, there exists a 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 $\mathcal{C}$. If $\mathcal{F}$ is of finite type, then there exists a 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. Proof omitted. $\square$


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