The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 54.64.1. Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$.

  1. There is an equivalence of categories

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

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

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

Proof. We observe that $\pi _1(X, \overline{x})$ is a profinite topological group, see Fundamental Groups, Definition 53.6.1. The left hand categories are defined in Section 54.63. The notation used in the right hand categories is taken from Fundamental Groups, Definition 53.2.1 for sets and Definition 54.56.1 for abelian groups. This explains the notation.

Assertion (1) follows from Lemma 54.63.4 and Fundamental Groups, Theorem 53.6.2. Parts (2) and (3) follow immediately from this by endowing the underlying (sheaves of) sets with additional structure. For example, a finite locally constant sheaf of abelian groups on $X_{\acute{e}tale}$ is the same thing as a finite locally constant sheaf of sets $\mathcal{F}$ together with a map $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ satisfying the usual axioms. The equivalence in (1) sends products to products and hence sends $+$ to an addition on the corresponding finite $\pi _1(X, \overline{x})$-set. Since $\pi _1(X, \overline{x})$-modules are the same thing as $\pi _1(X, \overline{x})$-sets with a compatible abelian group structure we obtain (2). Part (3) is proved in exactly the same way. $\square$


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