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

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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