Remark 59.53.2. Let $f : X \to S$ be a morphism of schemes. Let $K \in D(X_{\acute{e}tale})$. Let $\overline{s}$ be a geometric point of $S$. There are always canonical maps
\[ (Rf_*K)_{\overline{s}} \longrightarrow R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \longrightarrow R\Gamma (X_{\overline{s}}, K|_{X_{\overline{s}}}) \]
where $p : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \to X$ is the projection. Namely, consider the commutative diagram
\[ \xymatrix{ X_{\overline{s}} \ar[r] \ar[d]^{f_{\overline{s}}} & X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \ar[r]_-p \ar[d]^{f'} & X \ar[d]^ f \\ \overline{s} \ar[r]^-i & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \ar[r]^-j & S } \]
We have the base change maps
\[ i^{-1}Rf'_*(p^{-1}K) \to Rf_{\overline{s}, *}(K|_{X_{\overline{s}}}) \quad \text{and}\quad j^{-1}Rf_*K \to Rf'_*(p^{-1}K) \]
(Cohomology on Sites, Remark 21.19.3) for the two squares in this diagram. Taking global sections we obtain the desired maps. By Cohomology on Sites, Remark 21.19.5 the composition of these two maps is the usual (base change) map $(Rf_*K)_{\overline{s}} \to R\Gamma (X_{\overline{s}}, K|_{X_{\overline{s}}})$.
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