The Stacks project

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 29.54.11 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. $\square$


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