The Stacks project

Lemma 59.42.4. Let $f : X \to Y$ be a morphism of schemes. Denote $f_{small} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ the associated morphism of small étale topoi. Assume at least one of the following

  1. $f$ is integral, or

  2. $f$ is separated and locally quasi-finite.

Then the functor $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ has the following properties

  1. the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is always surjective,

  2. $f_{small, *}$ is faithful, and

  3. $f_{small, *}$ reflects injections and surjections.

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