Lemma 82.4.5. Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let $\overline{x} \to X$ be a geometric point. For any sheaf $\mathcal{F}$ on $Y_{\acute{e}tale}$ the canonical map

is bijective.

Lemma 82.4.5. Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let $\overline{x} \to X$ be a geometric point. For any sheaf $\mathcal{F}$ on $Y_{\acute{e}tale}$ the canonical map

\[ (f_*\mathcal{F})_{\overline{x}} \longrightarrow \Gamma (Y_{\overline{x}}, \mathcal{F}_{\overline{x}}) \]

is bijective.

**Proof.**
This is a special case of Lemma 82.4.4.
$\square$

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