The Stacks project

Lemma 63.3.10. Let $k$ be a field. Let $j : X \to \overline{X}$ be an open immersion of schemes over $k$ with $\overline{X}$ proper over $k$. For $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ there is a canonical isomorphism (see proof)

\[ H^0_ c(X, \mathcal{F}) \longrightarrow H^0_ c(\overline{X}, j_!\mathcal{F}) = H^0(\overline{X}, j_!\mathcal{F}) \]

where we have the equality on the right by Lemma 63.3.8.

Proof. We have $(j_!\mathcal{F})|_ X = \mathcal{F}$, see √Čtale Cohomology, Lemma 59.70.4. Thus the displayed arrow is the injective map $H^0_ c(X, \mathcal{G}|_ X) \to H^0_ c(\overline{X}, \mathcal{G})$ of Remark 63.3.9 for $\mathcal{G} = j_!\mathcal{F}$. The explicit nature of this map implies that it now suffices to show: if $s \in H^0(\overline{X}, \mathcal{G})$ is a section, then the support of $s$ is contained in the open $X$. 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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