The Stacks project

Lemma 63.4.5. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Then

  1. for $\mathcal{F}$ in $\textit{Ab}(X_{\acute{e}tale})$ and a geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ we have

    \[ (f_!\mathcal{F})_{\overline{y}} = \bigoplus \nolimits _{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \]

    functorially in $\mathcal{F}$, and

  2. the functor $f_! : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is exact and commutes with direct sums.

Proof. The formula for the stalks is immediate (and in fact equivalent) to Lemma 63.4.2. The exactness of the functor follows immediately from this and the fact that exactness may be checked on stalks, see Étale Cohomology, Theorem 59.29.10. $\square$

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