The Stacks project

Lemma 59.44.5. Let $f : X \to Y$ be a morphism of schemes. Assume that $f$ is universally injective and integral (for example a closed immersion). Then

  1. $f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ reflects injections and surjections,

  2. $f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ commutes with pushouts and coequalizers (and more generally finite connected colimits),

  3. $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves),

  4. the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any sheaf (of sets or of abelian groups) $\mathcal{F}$ on $X_{\acute{e}tale}$,

  5. the functor $f_{small, *}$ is faithful (on sheaves of sets and on abelian sheaves),

  6. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is exact, and

  7. the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ is almost cocontinuous.

Proof. By Lemmas 59.42.3, 59.43.4 and 59.44.4 we know that the morphism $f$ has properties (A), (B), and (C). Moreover, by Lemma 59.44.3 we know that the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$ is almost cocontinuous. Now we have

  1. property (C) implies (1) by Lemma 59.44.1,

  2. almost continuous implies (2) by Sites, Lemma 7.42.6,

  3. property (B) implies (3) by Lemma 59.43.1.

Properties (4), (5), and (6) follow formally from the first three, see Sites, Lemma 7.41.1 and Modules on Sites, Lemma 18.15.2. Property (7) we saw above. $\square$

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