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
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.
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}.
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
see Lemma 61.19.3 for the left hand side. By Lemma 61.19.3 we have
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
which follows once more from Étale Cohomology, Lemma 59.51.5. \square
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