The Stacks project

Lemma 59.75.7. Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $K \in D^+(X_{\acute{e}tale})$. Let $\overline{s}$ be a geometric point of $S$ and let $\overline{t}$ be a geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. We have a commutative diagram

\[ \xymatrix{ (Rf_*K)_{\overline{s}} \ar[r]_{sp} \ar@{=}[d] & (Rf_*K)_{\overline{t}} \ar@{=}[d] \\ R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), K) \ar[r] & R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}), K) } \]

where the bottom horizontal arrow arises as pullback by the morphism $\text{id}_ X \times c$ where $c : \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{S}})$ is the morphism introduced in Remark 59.75.2. The vertical arrows are given by Theorem 59.53.1.

Proof. This follows immediately from the description of $sp$ in Remark 59.75.2. $\square$

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