The Stacks project

Lemma 59.44.3. Let $f : X \to Y$ be a morphism of schemes. Assume that for any $V \to Y$ étale we have that

  1. $X \times _ Y V \to V$ has property (C), and

  2. $X \times _ Y V \to V$ is closed.

Then the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ is almost cocontinuous, see Sites, Definition 7.42.3.

Proof. Let $V \to Y$ be an object of $Y_{\acute{e}tale}$ and let $\{ U_ i \to X \times _ Y V\} _{i \in I}$ be a covering of $X_{\acute{e}tale}$. By assumption (1) for each $i$ we can find an étale morphism $h_ i : V_ i \to V$ and a surjective morphism $X \times _ Y V_ i \to U_ i$ over $X \times _ Y V$. Note that $\bigcup h_ i(V_ i) \subset V$ is an open set containing the closed set $Z = \mathop{\mathrm{Im}}(X \times _ Y V \to V)$. Let $h_0 : V_0 = V \setminus Z \to V$ be the open immersion. It is clear that $\{ V_ i \to V\} _{i \in I \cup \{ 0\} }$ is an étale covering such that for each $i \in I \cup \{ 0\} $ we have either $V_ i \times _ Y X = \emptyset $ (namely if $i = 0$), or $V_ i \times _ Y X \to V \times _ Y X$ factors through $U_ i \to X \times _ Y V$ (if $i \not= 0$). Hence the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$ is almost cocontinuous. $\square$

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