The Stacks project

Lemma 57.87.4. Let $f : X \to S$ be a proper morphism of schemes. Let $\overline{s} \to S$ be a geometric point. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ the canonical map

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

is bijective.

Proof. By Theorem 57.52.1 (for sheaves of sets) we have

\[ (f_*\mathcal{F})_{\overline{s}} = \Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p_{small}^{-1}\mathcal{F}) \]

where $p : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \to X$ is the projection. Since the residue field of the strictly henselian local ring $\mathcal{O}_{S, \overline{s}}^{sh}$ is $\kappa (s)^{sep}$ we conclude from the discussion above the lemma and Lemma 57.87.3. $\square$


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