The Stacks project

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

  1. $H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \mathop{\mathrm{Spec}}(K), \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \mathop{\mathrm{Spec}}(K), \mathcal{F})$

  2. $H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \mathop{\mathrm{Spec}}(K), \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(K), \mathcal{F})$

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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