The Stacks project

Proof. By Lemma 61.12.2 any pro-étale covering is an fpqc covering. Hence the formula defines a sheaf on $X_{pro\text{-}\acute{e}tale}$ by Étale Cohomology, Lemma 59.39.2. Let $a : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale})$ be the functor sending $\mathcal{F}$ to the sheaf given by the formula in the lemma. To show that $a = \epsilon ^{-1}$ it suffices to show that $a$ is a left adjoint to $\epsilon _*$.

Let $\mathcal{G}$ be an object of $\mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale})$. Recall that $\epsilon _*\mathcal{G}$ is simply given by the restriction of $\mathcal{G}$ to the full subcategory $X_{\acute{e}tale}$. Let $f : Y \to X$ be an object of $X_{pro\text{-}\acute{e}tale}$. We view $Y_{\acute{e}tale}$ as a subcategory of $X_{pro\text{-}\acute{e}tale}$. The restriction maps of the sheaf $\mathcal{G}$ define a map

Namely, for $U$ in $X_{\acute{e}tale}$ the value of $f_{{\acute{e}tale}, *}(\mathcal{G}|_{Y_{\acute{e}tale}})$ on $U$ is $\mathcal{G}(Y \times _ X U)$ and there is a restriction map $\mathcal{G}(U) \to \mathcal{G}(Y \times _ X U)$. By adjunction this determines a map

Putting these together for all $f : Y \to X$ in $X_{pro\text{-}\acute{e}tale}$ we obtain a canonical map $a(\epsilon _*\mathcal{G}) \to \mathcal{G}$.

Let $\mathcal{F}$ be an object of $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$. It is immediately clear that $\mathcal{F} = \epsilon _*a(\mathcal{F})$.

We claim the maps $\mathcal{F} \to \epsilon _*a(\mathcal{F})$ and $a(\epsilon _*\mathcal{G}) \to \mathcal{G}$ are the unit and counit of the adjunction (see Categories, Section 4.24). To see this it suffices to show that the corresponding maps

and

are mutually inverse. We omit the detailed verification. $\square$

Proof. Follows immediately from the description of $\epsilon ^{-1}$ in Lemma 61.19.1. $\square$

Proof. By the description of $\epsilon ^{-1}\mathcal{F}$ in Lemma 61.19.1, the displayed formula is a special case of Étale Cohomology, Theorem 59.51.3. (When $X$, $Y$, and the $Y_ i$ are all affine, see the easier to parse Étale Cohomology, Lemma 59.51.5.) The final statement follows immediately from this and Lemma 61.19.2. $\square$

Proof. We are going to use Cohomology on Sites, Lemma 21.10.9 to prove this. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{pro\text{-}\acute{e}tale})$ be the set of affine objects $U$ of $X_{pro\text{-}\acute{e}tale}$ such that $\mathcal{O}(X) \to \mathcal{O}(U)$ is ind-étale. Let $\text{Cov}$ be the set of pro-étale coverings $\{ U_ i \to U\} _{i = 1, \ldots , n}$ with $U \in \mathcal{B}$ such that $\mathcal{O}(U) \to \mathcal{O}(U_ i)$ is ind-étale for $i = 1, \ldots , n$. Properties (1) and (2) of Cohomology on Sites, Lemma 21.10.9 hold for $\mathcal{B}$ and $\text{Cov}$ by Lemmas 61.7.3, 61.7.2, and 61.12.5 and Proposition 61.9.1.

To check condition (3) suppose that $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ is an element of $\text{Cov}$. We have to show that the higher Cech cohomology groups of $\epsilon ^{-1}\mathcal{I}$ with respect to $\mathcal{U}$ are zero. First we write $U_ i = \mathop{\mathrm{lim}}\nolimits _{a \in A_ i} U_{i, a}$ as a directed inverse limit with $U_{i, a} \to U$ étale and $U_{i, a}$ affine. We think of $A_1 \times \ldots \times A_ n$ as a direct set with ordering $(a_1, \ldots , a_ n) \geq (a_1', \ldots , a_ n')$ if and only if $a_ i \geq a_ i'$ for $i = 1, \ldots , n$. Observe that $\mathcal{U}_{(a_1, \ldots , a_ n)} = \{ U_{i, a_ i} \to U\} _{i = 1, \ldots , n}$ is an étale covering for all $a_1, \ldots , a_ n \in A_1 \times \ldots \times A_ n$. Observe that

for all $i_0, \ldots , i_ p \in \{ 1, \ldots , n\} $ because limits commute with fibred products. Hence by Lemma 61.19.3 and exactness of filtered colimits we have

Thus it suffices to prove the vanishing for étale coverings of $U$!

Let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ be an étale covering with $U_ i$ affine. Write $U = \mathop{\mathrm{lim}}\nolimits _{b \in B} U_ b$ as a directed inverse limit with $U_ b$ affine and $U_ b \to X$ étale. By Limits, Lemmas 32.10.1, 32.4.13, and 32.8.10 we can choose a $b_0 \in B$ such that for $i = 1, \ldots , n$ there is an étale morphism $U_{i, b_0} \to U_{b_0}$ of affines such that $U_ i = U \times _{U_{b_0}} U_{i, b_0}$. Set $U_{i, b} = U_ b \times _{U_{b_0}} U_{i, b_0}$ for $b \geq b_0$. For $b$ large enough the family $\mathcal{U}_ b = \{ U_{i, b} \to U_ b\} _{i = 1, \ldots , n}$ is an étale covering, see Limits, Lemma 32.8.15. Exactly as before we find that

the final equality by Lemma 61.19.2. Since each of the Čech complexes on the right hand side is acyclic in positive degrees (Cohomology on Sites, Lemma 21.10.2) it follows that the one on the left is too. This proves condition (3) of Cohomology on Sites, Lemma 21.10.9. Since $X \in \mathcal{B}$ the lemma follows. $\square$

Proof. Let $\mathcal{I}$ be an injective abelian sheaf on $X_{\acute{e}tale}$. Recall that $R^ q\epsilon _*(\epsilon ^{-1}\mathcal{I})$ is the sheaf associated to $U \mapsto H^ q(U_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}\mathcal{I})$, see Cohomology on Sites, Lemma 21.7.4. By Lemma 61.19.4 we see that this is zero for $q > 0$ and $U$ affine and étale over $X$. Since every object of $X_{\acute{e}tale}$ has a covering by affine objects, it follows that $R^ q\epsilon _*(\epsilon ^{-1}\mathcal{I}) = 0$ for $q > 0$.

Let $K \in D^+(X_{\acute{e}tale})$. Choose a bounded below complex $\mathcal{I}^\bullet $ of injective abelian sheaves on $X_{\acute{e}tale}$ representing $K$. Then $\epsilon ^{-1}K$ is represented by $\epsilon ^{-1}\mathcal{I}^\bullet $. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) we see that $R\epsilon _*\epsilon ^{-1}K$ is represented by $\epsilon _*\epsilon ^{-1}\mathcal{I}^\bullet $. By Lemma 61.19.2 we conclude that $R\epsilon _*\epsilon ^{-1}\mathcal{I}^\bullet = \mathcal{I}^\bullet $ and the proof of (2) is complete. Part (1) is a special case of (2). $\square$

Proof. Immediate consequence of Lemma 61.19.5 and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). $\square$

Proof. Since the functor $\epsilon ^{-1}$ is fully faithful by Lemma 61.19.2 it is clear that the map $H^1(X_{\acute{e}tale}, \mathcal{G}) \to H^1(X_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}\mathcal{G})$ is injective. To show surjectivity it suffices to show that any $\epsilon ^{-1}\mathcal{G}$-torsor $\mathcal{F}$ is étale locally trivial. To do this we may assume that $X$ is affine. Thus we reduce to proving surjectivity for $X$ affine.

Choose a covering $\{ U \to X\} $ with (a) $U$ affine, (b) $\mathcal{O}(X) \to \mathcal{O}(U)$ ind-étale, and (c) $\mathcal{F}(U)$ nonempty. We can do this by Proposition 61.9.1 and the fact that standard pro-étale coverings of $X$ are cofinal among all pro-étale coverings of $X$ (Lemma 61.12.5). Write $U = \mathop{\mathrm{lim}}\nolimits U_ i$ as a limit of affine schemes étale over $X$. Pick $s \in \mathcal{F}(U)$. Let $g \in \epsilon ^{-1}\mathcal{G}(U \times _ X U)$ be the unique section such that $g \cdot \text{pr}_1^*s = \text{pr}_2^*s$ in $\mathcal{F}(U \times _ X U)$. Then $g$ satisfies the cocycle condition

in $\epsilon ^{-1}\mathcal{G}(U \times _ X U \times _ X U)$. By Lemma 61.19.3 we have

and

hence we can find an $i$ and an element $g_ i \in \mathcal{G}(U_ i \times _ X U_ i)$ mapping to $g$ satisfying the cocycle condition. The cocycle $g_ i$ then defines a torsor for $\mathcal{G}$ on $X_{\acute{e}tale}$ whose pullback is isomorphic to $\mathcal{F}$ by construction. Some details omitted (namely, the relationship between torsors and 1-cocycles which should be added to the chapter on cohomology on sites). $\square$

Proof. To prove (1) we will prove conditions (1) – (4) of Homology, Lemma 12.10.3. Since $\epsilon ^{-1}$ is fully faithful (Lemma 61.19.2) and exact, everything is clear except for condition (4). However, if

is a short exact sequence of sheaves of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$, then we get

which by Lemma 61.19.5 is the same as a short exact sequence

Pulling pack we find that $\mathcal{G} = \epsilon ^{-1}\epsilon _*\mathcal{G}$. This proves (1).

Part (2) follows from part (1) and Cohomology on Sites, Lemma 21.28.5. $\square$

Proof. Let $\mathcal{F}$ be a locally constant sheaf of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$ of finite presentation. Choose a pro-étale covering $\{ U_ i \to X\} $ such that $\mathcal{F}|_{U_ i}$ is constant, say $\mathcal{F}|_{U_ i} \cong \underline{M_ i}_{U_ i}$. Observe that $U_ i \times _ X U_ j$ is empty if $M_ i$ is not isomorphic to $M_ j$. For each $\Lambda $-module $M$ let $I_ M = \{ i \in I \mid M_ i \cong M\} $. As pro-étale coverings are fpqc coverings and by Descent, Lemma 35.13.6 we see that $U_ M = \bigcup _{i \in I_ M} \mathop{\mathrm{Im}}(U_ i \to X)$ is an open subset of $X$. Then $X = \coprod U_ M$ is a disjoint open covering of $X$. We may replace $X$ by $U_ M$ for some $M$ and assume that $M_ i = M$ for all $i$.

Consider the sheaf $\mathcal{I} = \mathit{Isom}(\underline{M}, \mathcal{F})$. This sheaf is a torsor for $\mathcal{G} = \mathit{Isom}(\underline{M}, \underline{M})$. By Modules on Sites, Lemma 18.43.4 we have $\mathcal{G} = \underline{G}$ where $G = \mathit{Isom}_\Lambda (M, M)$. Since torsors for the étale topology and the pro-étale topology agree by Lemma 61.19.7 it follows that $\mathcal{I}$ has sections étale locally on $X$. Thus $\mathcal{F}$ is étale locally a constant sheaf which is what we had to show. $\square$

Proof. The categories $D_{flc}(X_{\acute{e}tale}, \Lambda )$ and $D_{flc}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ are strictly full, saturated, triangulated subcategories of $D(X_{\acute{e}tale}, \Lambda )$ and $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ by Modules on Sites, Lemma 18.43.5 and Derived Categories, Section 13.17. The statement of the lemma follows by combining Lemmas 61.19.8 and 61.19.9. $\square$

Proof. Assertion (2) for $K_ n$ follows from the more general Cohomology on Sites, Lemma 21.46.9. Assertion (1) for $K_ n$ follows by induction on $n$ from the distinguished triangles

and the isomorphism

and the fact proven in Lemma 61.19.8 that the essential image of $\epsilon ^{-1}$ is a triangulated subcategory of $D^+(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$. $\square$