Lemma 61.19.9. Let $X$ be a scheme. Let $\Lambda $ be a ring. The functor $\epsilon ^{-1}$ defines an equivalence of categories
\[ \left\{ \begin{matrix} \text{locally constant sheaves}
\\ \text{of }\Lambda \text{-modules on }X_{\acute{e}tale}
\\ \text{of finite presentation}
\end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{locally constant sheaves}
\\ \text{of }\Lambda \text{-modules on }X_{pro\text{-}\acute{e}tale}
\\ \text{of finite presentation}
\end{matrix} \right\} \]
Proof.
Let $\mathcal{F}$ be a locally constant sheaf of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$ of finite presentation. Choose a pro-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 pro-étale coverings are fpqc coverings and by Descent, Lemma 35.13.6 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 $X$. We may replace $X$ by $U_ M$ for some $M$ and assume that $M_ i = M$ for all $i$.
Consider the sheaf $\mathcal{I} = \mathit{Isom}(\underline{M}, \mathcal{F})$. This sheaf is a torsor for $\mathcal{G} = \mathit{Isom}(\underline{M}, \underline{M})$. By Modules on Sites, Lemma 18.43.4 we have $\mathcal{G} = \underline{G}$ where $G = \mathit{Isom}_\Lambda (M, M)$. Since torsors for the étale topology and the pro-étale topology agree by Lemma 61.19.7 it follows that $\mathcal{I}$ has sections étale locally on $X$. Thus $\mathcal{F}$ is étale locally a constant sheaf which is what we had to show.
$\square$
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