The Stacks project

Lemma 60.19.1. 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 58.34.

Proof. By Lemma 60.12.2 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 58.39.2. 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. $\square$


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