The Stacks project

Lemma 62.4.1. Let $f : X \to Y$ be a separated and locally quasi-finite morphism of schemes. Functorially in $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ there is a canonical isomorphism(!)

\[ f_{p!}\mathcal{F} \longrightarrow f_!\mathcal{F} \]

of abelian presheaves which identifies the sheaf $f_!\mathcal{F}$ of Definition 62.3.3 with the presheaf $f_{p!}\mathcal{F}$ constructed above.

Proof. Let $V$ be an object of $Y_{\acute{e}tale}$. If $Z \subset X_ V$ is locally closed and finite over $V$, then, since $f$ is separated, we see that the morphism $Z \to X_ V$ is a closed immersion. Moreover, if $Z_ i$, $i = 1, \ldots , n$ are closed subschemes of $X_ V$ finite over $V$, then $Z_1 \cup \ldots \cup Z_ n$ (scheme theoretic union) is a closed subscheme finite over $V$. Hence in this case the colimit ( defining $f_{p!}\mathcal{F}(V)$ is directed and we find that $f_{!p}\mathcal{F}(V)$ is simply equal to the set of sections of $\mathcal{F}(X_ V)$ whose support is finite over $V$. Since any closed subset of $X_ V$ which is proper over $V$ is actually finite over $V$ (as $f$ is locally quasi-finite) we conclude that this is equal to $f_!\mathcal{F}(V)$ by its very definition. $\square$

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