Lemma 59.65.1. Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$.

There is an equivalence of categories

\[ \left\{ \begin{matrix} \text{finite locally constant} \\ \text{sheaves of sets on }X_{\acute{e}tale} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{finite }\pi _1(X, \overline{x})\text{-sets} \end{matrix} \right\} \]There is an equivalence of categories

\[ \left\{ \begin{matrix} \text{finite locally constant} \\ \text{sheaves of abelian groups on }X_{\acute{e}tale} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{finite }\pi _1(X, \overline{x})\text{-modules} \end{matrix} \right\} \]Let $\Lambda $ be a finite ring. There is an equivalence of categories

\[ \left\{ \begin{matrix} \text{finite type, locally constant} \\ \text{sheaves of }\Lambda \text{-modules on }X_{\acute{e}tale} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{finite }\pi _1(X, \overline{x})\text{-modules endowed} \\ \text{with commuting }\Lambda \text{-module structure} \end{matrix} \right\} \]

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