The Stacks project

Lemma 59.43.5. Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is integral (for example finite). Then

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

2. $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,

3. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is faithful and reflects injections and surjections, and

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