Definition 61.27.1. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. A sheaf of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$ is constructible if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod _ i U_ i$ such that $\mathcal{F}|_{U_ i}$ is of finite type and locally constant for all $i$.
61.27 Constructible sheaves on the pro-étale site
We stick to constructible sheaves of $\Lambda $-modules for a Noetherian ring. In the future we intend to discuss constructible sheaves of sets, groups, etc.
Again this does not give anything “new”.
Lemma 61.27.2. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. The functor $\epsilon ^{-1}$ defines an equivalence of categories between constructible sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$ and constructible sheaves of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$.
\[ \left\{ \begin{matrix} \text{constructible sheaves of}
\\ \Lambda \text{-modules on }X_{\acute{e}tale}
\\ \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{constructible sheaves of}
\\ \Lambda \text{-modules on }X_{pro\text{-}\acute{e}tale}
\\ \end{matrix} \right\} \]
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. $\square$
Lemma 61.27.3. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. The category of constructible sheaves of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$ is a weak Serre subcategory of $\textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda )$.
Proof. This is a formal consequence of Lemmas 61.27.2 and 61.19.8 and the result for the étale site (Étale Cohomology, Lemma 59.71.6). $\square$
Lemma 61.27.4. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $D_ c(X_{\acute{e}tale}, \Lambda )$, resp. $D_ c(X_{pro\text{-}\acute{e}tale}, \Lambda )$ be the full subcategory of $D(X_{\acute{e}tale}, \Lambda )$, resp. $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ consisting of those complexes whose cohomology sheaves are constructible sheaves of $\Lambda $-modules. Then is an equivalence of categories.
\[ \epsilon ^{-1} : D_ c^+(X_{\acute{e}tale}, \Lambda ) \longrightarrow D_ c^+(X_{pro\text{-}\acute{e}tale}, \Lambda ) \]
Proof. The categories $D_ c(X_{\acute{e}tale}, \Lambda )$ and $D_ c(X_{pro\text{-}\acute{e}tale}, \Lambda )$ are strictly full, saturated, triangulated subcategories of $D(X_{\acute{e}tale}, \Lambda )$ and $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ by Étale Cohomology, Lemma 59.71.6 and Lemma 61.27.3 and Derived Categories, Section 13.17. The statement of the lemma follows by combining Lemmas 61.19.8 and 61.27.2. $\square$
Lemma 61.27.5. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $K, L \in D_ c^-(X_{pro\text{-}\acute{e}tale}, \Lambda )$. Then $K \otimes _\Lambda ^\mathbf {L} L$ is in $D_ c^-(X_{pro\text{-}\acute{e}tale}, \Lambda )$.
Proof. Note that $H^ i(K \otimes _\Lambda ^\mathbf {L} L)$ is the same as $H^ i(\tau _{\geq i - 1}K \otimes _\Lambda ^\mathbf {L} \tau _{\geq i - 1}L)$. Thus we may assume $K$ and $L$ are bounded. In this case we can apply Lemma 61.27.4 to reduce to the case of the étale site, see Étale Cohomology, Lemma 59.76.6. $\square$
Lemma 61.27.6. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $K$ be an object of $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$. Set $K_ n = K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$. If $K_1$ is in $D^-_ c(X_{pro\text{-}\acute{e}tale}, \Lambda /I)$, then $K_ n$ is in $D^-_ c(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$ for all $n$.
Proof. Consider the distinguished triangles
\[ K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} \to K_{n + 1} \to K_ n \to K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}}[1] \]
and the isomorphisms
\[ K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} = K_1 \otimes _{\Lambda /I}^\mathbf {L} \underline{I^ n/I^{n + 1}} \]
By Lemma 61.27.5 we see that this tensor product has constructible cohomology sheaves (and vanishing when $K_1$ has vanishing cohomology). Hence by induction on $n$ using Lemma 61.27.3 we see that each $K_ n$ has constructible cohomology sheaves. $\square$
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