The Stacks project

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. $\square$

Comments (2)

Comment #8497 by Lukas Krinner on

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

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