The Stacks project

Lemma 59.64.8. Let $X$ be a connected scheme. Let $\Lambda $ be a ring and let $\mathcal{F}$ be a locally constant sheaf of $\Lambda $-modules. Then there exists a $\Lambda $-module $M$ and an étale covering $\{ U_ i \to X\} $ such that $\mathcal{F}|_{U_ i} \cong \underline{M}|_{U_ i}$.

Proof. Choose an étale covering $\{ U_ i \to X\} $ such that $\mathcal{F}|_{U_ i}$ is constant, say $\mathcal{F}|_{U_ i} \cong \underline{M_ i}_{U_ i}$. Observe that $U_ i \times _ X U_ j$ is empty if $M_ i$ is not isomorphic to $M_ j$. For each $\Lambda $-module $M$ let $I_ M = \{ i \in I \mid M_ i \cong M\} $. As étale morphisms are open we see that $U_ M = \bigcup _{i \in I_ M} \mathop{\mathrm{Im}}(U_ i \to X)$ is an open subset of $X$. Then $X = \coprod U_ M$ is a disjoint open covering of $X$. As $X$ is connected only one $U_ M$ is nonempty and the lemma follows. $\square$

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