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59.65 Locally constant sheaves and the fundamental group

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

Lemma 59.65.1. 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 58.6.1. The left hand categories are defined in Section 59.64. The notation used in the right hand categories is taken from Fundamental Groups, Definition 58.2.1 for sets and Definition 59.57.1 for abelian groups. This explains the notation.

Assertion (1) follows from Lemma 59.64.4 and Fundamental Groups, Theorem 58.6.2. 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. $\square$

Lemma 59.65.2. Let $X$ be an irreducible, geometrically unibranch scheme. Let $\overline{x}$ be a geometric point of $X$. Let $\Lambda $ be a ring. There is an equivalence of categories

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

Proof. The proof given in Lemma 59.65.1 does not work as a finite $\Lambda $-module $M$ may not have a finite underlying set.

Let $\nu : X^\nu \to X$ be the normalization morphism. By Morphisms, Lemma 29.54.12 this is a universal homeomorphism. By Fundamental Groups, Proposition 58.8.4 this induces an isomorphism $\pi _1(X^\nu , \overline{x}) \to \pi _1(X, \overline{x})$ and by Theorem 59.45.2 we get an equivalence of category between finite type, locally constant $\Lambda $-modules on $X_{\acute{e}tale}$ and on $X^\nu _{\acute{e}tale}$. This reduces us to the case where $X$ is an integral normal scheme.

Assume $X$ is an integral normal scheme. Let $\eta \in X$ be the generic point. Let $\overline{\eta }$ be a geometric point lying over $\eta $. By Fundamental Groups, Proposition 58.11.3 have a continuous surjection

\[ \text{Gal}(\kappa (\eta )^{sep}/\kappa (\eta )) = \pi _1(\eta , \overline{\eta }) \longrightarrow \pi _1(X, \overline{\eta }) \]

whose kernel is described in Fundamental Groups, Lemma 58.13.2. Let $\mathcal{F}$ be a finite type, locally constant sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$. Let $M = \mathcal{F}_{\overline{\eta }}$ be the stalk of $\mathcal{F}$ at $\overline{\eta }$. We obtain a continuous action of $\text{Gal}(\kappa (\eta )^{sep}/\kappa (\eta ))$ on $M$ by Section 59.56. Our goal is to show that this action factors through the displayed surjection. Since $\mathcal{F}$ is of finite type, $M$ is a finite $\Lambda $-module. Since $\mathcal{F}$ is locally constant, for every $x \in X$ the restriction of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^{sh})$ is constant. Hence the action of $\text{Gal}(K^{sep}/K_ x^{sh})$ (with notation as in Fundamental Groups, Lemma 58.13.2) on $M$ is trivial. We conclude we have the factorization as desired.

On the other hand, suppose we have a finite $\Lambda $-module $M$ with a continuous action of $\pi _1(X, \overline{\eta })$. We are going to construct an $\mathcal{F}$ such that $M \cong \mathcal{F}_{\overline{\eta }}$ as $\Lambda [\pi _1(X, \overline{\eta })]$-modules. Choose generators $m_1, \ldots , m_ r \in M$. Since the action of $\pi _1(X, \overline{\eta })$ on $M$ is continuous, for each $i$ there exists an open subgroup $N_ i$ of the profinite group $\pi _1(X, \overline{\eta })$ such that every $\gamma \in H_ i$ fixes $m_ i$. We conclude that every element of the open subgroup $H = \bigcap _{i = 1, \ldots , r} H_ i$ fixes every element of $M$. After shrinking $H$ we may assume $H$ is an open normal subgroup of $\pi _1(X, \overline{\eta })$. Set $G = \pi _1(X, \overline{\eta })/H$. Let $f : Y \to X$ be the corresponding Galois finite étale $G$-cover. We can view $f_*\underline{\mathbf{Z}}$ as a sheaf of $\mathbf{Z}[G]$-modules on $X_{\acute{e}tale}$. Then we just take

\[ \mathcal{F} = f_*\underline{\mathbf{Z}} \otimes _{\underline{\mathbf{Z}[G]}} \underline{M} \]

We leave it to the reader to compute $\mathcal{F}_{\overline{\eta }}$. We also omit the verification that this construction is the inverse to the construction in the previous paragraph. $\square$

Remark 59.65.3. The equivalences of Lemmas 59.65.1 and 59.65.2 are compatible with pullbacks. For example, 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 58.6.2. A similar result holds for the other cases.

Comments (2)

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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