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58.64 Locally constant sheaves and the fundamental group

We can relate locally constant sheaves to the fundamental group of a scheme in some cases.

Lemma 58.64.1. 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 57.6.1. The left hand categories are defined in Section 58.63. The notation used in the right hand categories is taken from Fundamental Groups, Definition 57.2.1 for sets and Definition 58.56.1 for abelian groups. This explains the notation.

Assertion (1) follows from Lemma 58.63.4 and Fundamental Groups, Theorem 57.6.2. 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. $\square$

Remark 58.64.2. The equivalences of Lemma 58.64.1 are compatible with pullbacks. More precisely, suppose $f : Y \to X$ is a morphism of connected schemes. Let $\overline{y}$ be geometric point of $Y$ and set $\overline{x} = f(\overline{y})$. Then the diagram

\[ \xymatrix{ \text{finite locally constant sheaves of sets on }Y_{\acute{e}tale}\ar[r] & \text{finite }\pi _1(Y, \overline{y})\text{-sets} \\ \text{finite locally constant sheaves of sets on }X_{\acute{e}tale}\ar[r] \ar[u]_{f^{-1}} & \text{finite }\pi _1(X, \overline{x})\text{-sets} \ar[u] } \]

is commutative, where the vertical arrow on the right comes from the continuous homomorphism $\pi _1(Y, \overline{y}) \to \pi _1(X, \overline{x})$ induced by $f$. This follows immediately from the commutative diagram in Fundamental Groups, Theorem 57.6.2.

Comments (1)

Comment #5501 by Rex on

It seems that the proof here is implicitly using the equivalence between finite locally constant sheaves of sets on X and finite etale covers of X (as in Theorem 57.6.2). In which section is this equivalence written down?

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