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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