Definition 62.4.4. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. We define the direct image with compact support to be the functor

$f_! : \textit{Ab}(X_{\acute{e}tale}) \longrightarrow \textit{Ab}(Y_{\acute{e}tale})$

defined by the formula $f_!\mathcal{F} = (f_{p!}\mathcal{F})^\#$, i.e., $f_!\mathcal{F}$ is the sheafification of the presheaf $f_{p!}\mathcal{F}$ constructed above.

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