Lemma 61.19.1. With notation as above. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. The rule

$X_{pro\text{-}\acute{e}tale}\longrightarrow \textit{Sets},\quad (f : Y \to X) \longmapsto \Gamma (Y_{\acute{e}tale}, f_{\acute{e}tale}^{-1}\mathcal{F})$

is a sheaf and is equal to $\epsilon ^{-1}\mathcal{F}$. Here $f_{\acute{e}tale}: Y_{\acute{e}tale}\to X_{\acute{e}tale}$ is the morphism of small étale sites constructed in Étale Cohomology, Section 59.34.

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

$\epsilon _*\mathcal{G} = \mathcal{G}|_{X_{\acute{e}tale}} \longrightarrow f_{{\acute{e}tale}, *}(\mathcal{G}|_{Y_{\acute{e}tale}})$

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

$f_{\acute{e}tale}^{-1}(\epsilon _*\mathcal{G}) \to \mathcal{G}|_{Y_{\acute{e}tale}}$

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

$\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale})}(a(\mathcal{F}), \mathcal{G}) \to \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})}(\mathcal{F}, \epsilon ^{-1}\mathcal{G})$

and

$\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})}(\mathcal{F}, \epsilon ^{-1}\mathcal{G}) \to \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale})}(a(\mathcal{F}), \mathcal{G})$

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

Comment #6481 by Owen on

Thank you for catching the error in my purported proof of 61.19.3. This lemma is very nice and I agree it's good to have anyway. In the last two displayed equations (describing the adjunction), the $\epsilon^{-1}$ should be $\epsilon_*$.

To establish the adjunction, I find it helpful to describe the counit $a\epsilon_*\mathcal G\to\mathcal G$ on sections over $f:Y\to X$ in $X_{\text{proét}}$ as the natural map coming from restriction. Here the colimit is over those $X$-morphisms $b:Y\to U$ to some $U\to X$ étale. Then the only nontrivial assertion is that any map $a\mathcal F\to\mathcal G$ factors on sections over $f$ as This is true since $a\mathcal F\to\mathcal G$ is a morphism of sheaves.

Thanks again!

Comment #6554 by on

@#6481. Why does the formula $a\epsilon_* \mathcal{G}(U) = \text{colim}_b \mathcal{G}(U/X)$ hold? I think you need to sheafifiy?

Anyway, this lemma is a consequence of a "meta lemma" that we could formulate and prove that would imply not just this lemma but also Lemmas 59.39.2, 59.100.1, 59.102.1, 59.103.1, 61.19.1, and 83.5.1. It would be something about some subcategory of the category of schemes over $X$ endowed with a topology all of whose coverings are fpqc coverings and having at least all etale coverings in there...

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