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61.23 Functoriality of the pro-étale site

Let $f : X \to Y$ be a morphism of schemes. The functor $Y_{pro\text{-}\acute{e}tale}\to X_{pro\text{-}\acute{e}tale}$, $V \mapsto X \times _ Y V$ induces a morphism of sites $f_{pro\text{-}\acute{e}tale}: X_{pro\text{-}\acute{e}tale}\to Y_{pro\text{-}\acute{e}tale}$, see Sites, Proposition 7.14.7. In fact, we obtain a commutative diagram of morphisms of sites

\[ \xymatrix{ X_{pro\text{-}\acute{e}tale}\ar[r]_\epsilon \ar[d]_{f_{pro\text{-}\acute{e}tale}} & X_{\acute{e}tale}\ar[d]^{f_{\acute{e}tale}} \\ Y_{pro\text{-}\acute{e}tale}\ar[r]^\epsilon & Y_{\acute{e}tale}} \]

where $\epsilon $ is as in Section 61.19. In particular we have $\epsilon ^{-1} f_{\acute{e}tale}^{-1} = f_{pro\text{-}\acute{e}tale}^{-1} \epsilon ^{-1}$. Here is the corresponding result for pushforward.

Lemma 61.23.1. Let $f : X \to Y$ be a morphism of schemes.

  1. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$. Then we have $f_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F} = \epsilon ^{-1}f_{{\acute{e}tale}, *}\mathcal{F}$.

  2. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Then we have $Rf_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F} = \epsilon ^{-1}Rf_{{\acute{e}tale}, *}\mathcal{F}$.

Proof. Proof of (1). Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$. There is a canonical map $\epsilon ^{-1}f_{{\acute{e}tale}, *}\mathcal{F} \to f_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F}$, see Sites, Section 7.45. To show it is an isomorphism we may work (Zariski) locally on $Y$, hence we may assume $Y$ is affine. In this case every object of $Y_{pro\text{-}\acute{e}tale}$ has a covering by objects $V = \mathop{\mathrm{lim}}\nolimits V_ i$ which are limits of affine schemes $V_ i$ étale over $Y$ (by Proposition 61.9.1 for example). Evaluating the map $\epsilon ^{-1}f_{{\acute{e}tale}, *}\mathcal{F} \to f_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F}$ on $V$ we obtain a map

\[ \mathop{\mathrm{colim}}\nolimits \Gamma (X \times _ Y V_ i, \mathcal{F}) \longrightarrow \Gamma (X \times _ Y V, \epsilon ^*\mathcal{F}). \]

see Lemma 61.19.3 for the left hand side. By Lemma 61.19.3 we have

\[ \Gamma (X \times _ Y V, \epsilon ^*\mathcal{F}) = \Gamma (X \times _ Y V, \mathcal{F}) \]

Hence the result holds by Étale Cohomology, Lemma 59.51.5.

Proof of (2). Arguing in exactly the same manner as above we see that it suffices to show that

\[ \mathop{\mathrm{colim}}\nolimits H^ i_{\acute{e}tale}(X \times _ Y V_ i, \mathcal{F}) \longrightarrow H^ i_{\acute{e}tale}(X \times _ Y V, \mathcal{F}) \]

which follows once more from Étale Cohomology, Lemma 59.51.5. $\square$

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