Lemma 63.3.6. Let Y be a scheme. Let j : X \to \overline{X} be an open immersion of schemes over Y with \overline{X} proper over Y. Denote f : X \to Y and \overline{f} : \overline{X} \to Y the structure morphisms. For \mathcal{F} \in \textit{Ab}(X_{\acute{e}tale}) there is a canonical isomorphism (see proof)
f_!\mathcal{F} \longrightarrow \overline{f}_!j_!\mathcal{F}
As we have \overline{f}_! = \overline{f}_* by Lemma 63.3.4 we obtain \overline{f}_* \circ j_! = f_! as functors \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale}).
Proof.
We have (j_!\mathcal{F})|_ X = \mathcal{F}, see Étale Cohomology, Lemma 59.70.4. Thus the displayed arrow is the injective map f_!(\mathcal{G}|_ X) \to \overline{f}_!\mathcal{G} of Remark 63.3.5 for \mathcal{G} = j_!\mathcal{F}. The explicit nature of this map implies that it now suffices to show: if V \in Y_{\acute{e}tale} and s \in \overline{f}_!\mathcal{G}(V) = \overline{f}_*\mathcal{G}(V) = \mathcal{G}(\overline{X}_ V) is a section, then the support of s is contained in the open X_ V \subset \overline{X}_ V. This is immediate from the fact that the stalks of \mathcal{G} are zero at geometric points of \overline{X} \setminus X.
\square
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