Lemma 61.27.2. Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. The functor $\epsilon ^{-1}$ defines an equivalence of categories

$\left\{ \begin{matrix} \text{constructible sheaves of} \\ \Lambda \text{-modules on }X_{\acute{e}tale} \\ \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{constructible sheaves of} \\ \Lambda \text{-modules on }X_{pro\text{-}\acute{e}tale} \\ \end{matrix} \right\}$

between constructible sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$ and constructible sheaves of $\Lambda$-modules on $X_{pro\text{-}\acute{e}tale}$.

Proof. By Lemma 61.19.2 the functor $\epsilon ^{-1}$ is fully faithful and commutes with pullback (restriction) to the strata. Hence $\epsilon ^{-1}$ of a constructible étale sheaf is a constructible pro-étale sheaf. To finish the proof let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_{pro\text{-}\acute{e}tale}$ as in Definition 61.27.1. There is a canonical map

$\epsilon ^{-1}\epsilon _*\mathcal{F} \longrightarrow \mathcal{F}$

We will show this map is an isomorphism. This will prove that $\mathcal{F}$ is in the essential image of $\epsilon ^{-1}$ and finish the proof (details omitted).

Since it suffices to prove this locally on $X$ we may assume $X$ is quasi-compact and quasi-separated and that we have a finite partition $X = \coprod _{i = 1, \ldots , n} X_ i$ by constructible locally closed strata such that $\mathcal{F}|_{X_ i}$ is locally constant of finite type. We will use induction on $n$. The base case $n = 1$ follows from Lemma 61.19.9. Take a point $x \in X$; then $x \in X_ k$ for some $k$. It suffices to show the displayed map is an isomorphism in an open neighbourhood of $x$. Hence we may assume $X_ k$ is closed in $X$. Set $Z = X_ k$ and denote $U \subset X$ the complement of $Z$. Observe that the induction hypothesis applies to the restriction of $\mathcal{F}$ to $Z$ and to $U$. By Lemma 61.26.5 we have a short exact sequence

$0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0$

on $X_{pro\text{-}\acute{e}tale}$. Functoriality gives a commutative diagram

$\xymatrix{ 0 \ar[r] & \epsilon ^{-1}\epsilon _*j_!j^{-1}\mathcal{F} \ar[r] \ar[d] & \epsilon ^{-1}\epsilon _*\mathcal{F} \ar[r] \ar[d] & \epsilon ^{-1}\epsilon _*i_*i^{-1}\mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & j_!j^{-1}\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & i_*i^{-1}\mathcal{F} \ar[r] & 0 }$

By induction we know that on the one hand $\epsilon ^{-1}\epsilon _*i^{-1}\mathcal{F} \to i^{-1}\mathcal{F}$ and $\epsilon ^{-1}\epsilon _*j^{-1}\mathcal{F} \to j^{-1}\mathcal{F}$ are isomorphisms and on the other that $i^{-1}\mathcal{F} = \epsilon ^{-1}\mathcal{A}$ and $j^{-1}\mathcal{F} = \epsilon ^{-1}\mathcal{B}$ for some constructible sheaves of $\Lambda$-modules $\mathcal{A}$ on $Z_{\acute{e}tale}$ and $\mathcal{B}$ on $U_{\acute{e}tale}$. Then

$\epsilon ^{-1}\epsilon _*j_!j^{-1}\mathcal{F} = \epsilon ^{-1}\epsilon _*j_!\epsilon ^{-1}\mathcal{B} = \epsilon ^{-1}\epsilon _*\epsilon ^{-1}j_!\mathcal{B} = \epsilon ^{-1}j_!\mathcal{B} = j_!\epsilon ^{-1}\mathcal{B} = j_!j^{-1}\mathcal{F}$

the second equality by Lemma 61.26.2, the third equality by Lemma 61.19.2, and the fourth equality by Lemma 61.26.2 again. Similarly, we have

$\epsilon ^{-1}\epsilon _*i_*i^{-1}\mathcal{F} = \epsilon ^{-1}\epsilon _*i_*\epsilon ^{-1}\mathcal{A} = \epsilon ^{-1}\epsilon _*\epsilon ^{-1}i_*\mathcal{A} = \epsilon ^{-1}i_*\mathcal{A} = i_*\epsilon ^{-1}\mathcal{A} = i_*i^{-1}\mathcal{F}$

this time using Lemma 61.23.1. By the five lemma we conclude the vertical map in the middle of the big diagram is an isomorphism. $\square$

Comment #8497 by Lukas Krinner on

Approximately in the middle of the proof, the following is stated: ''Let $U \subset X$ be one of the open strata in the partition." Why is one of the $X_i$ necessarily open? (We did not even assume that $X$ is noetherian.) Maybe the argument should be: Let $X_k \overset{i}{\hookrightarrow} U_k \overset{j}{\hookrightarrow} X$ be a composition, where $i$ is a closed immersion and $j$ an open immersion. It suffices to show the desired isomorphism on the $U_k$'s. Hence, we can assume $X_k$ is closed. Then define $Z:= X_k$ and continue with the proof as described above (using for $U$ the complement of $Z$ instead of ''one of the open strata").

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