The Stacks project

Lemma 62.3.6. 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 62.3.4 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 √Čtale Cohomology, Lemma 59.70.4. Thus the displayed arrow is the injective map $f_!(\mathcal{G}|_ X) \to \overline{f}_!\mathcal{G}$ of Remark 62.3.5 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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