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. $\square$

Comment #8521 by Julian Demeio on

Is finiteness needed here?

Comment #8522 by Julian Demeio on

*Sorry, I mean "finite type". Is it true also for general sheaves of Λ-modules?

Comment #9121 by on

No. The issue is that an infinite direct sum of locally constant sheaves needn't be locally constant, whereas that is the case for modules endowed with continuous action. An example would be to consider the direct sum of $\mu_n$ for all $n$ on $\text{Spec}(\mathbf{Q})$. There is no single etale covering that trivializes this thing.

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