Definition 63.4.4. 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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