Lemma 59.86.6. Let f : X \to S be a morphism of schemes. Let \overline{x} be a geometric point of X with image \overline{s} in S. Let \mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) be a morphism with K a separably closed field. Let \mathcal{F} be an abelian sheaf on \mathop{\mathrm{Spec}}(K)_{\acute{e}tale}. Let q \geq 0. The following are equivalent
H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \mathop{\mathrm{Spec}}(K), \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \mathop{\mathrm{Spec}}(K), \mathcal{F})
H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \mathop{\mathrm{Spec}}(K), \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(K), \mathcal{F})
Proof.
Observe that \mathop{\mathrm{Spec}}(K) \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) is the spectrum of a filtered colimit of étale algebras over K. Since K is separably closed, each étale K-algebra is a finite product of copies of K. Thus we can write
\mathop{\mathrm{Spec}}(K) \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) = \mathop{\mathrm{lim}}\nolimits _{i \in I} \coprod \nolimits _{a \in A_ i} \mathop{\mathrm{Spec}}(K)
as a cofiltered limit where each term is a disjoint union of copies of \mathop{\mathrm{Spec}}(K) over a finite set A_ i. Note that A_ i is nonempty as we are given \mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}). It follows that
\begin{align*} \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \mathop{\mathrm{Spec}}(K) & = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \left( \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \mathop{\mathrm{Spec}}(K)\right) \\ & = \mathop{\mathrm{lim}}\nolimits _{i \in I} \coprod \nolimits _{a \in A_ i} \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \mathop{\mathrm{Spec}}(K) \end{align*}
Since taking cohomology in our setting commutes with limits of schemes (Theorem 59.51.3) we conclude.
\square
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