**Proof.**
Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}[n]$ where the colimit is over the multiplicative system of integers invertible in $K$. Since cohomology commutes with filtered colimits in our situation (for a precise reference see Lemma 54.80.3), it suffices to prove the lemma for $\mathcal{F}[n]$. Thus we may assume $\mathcal{F}$ is a sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules for some $n$ invertible in $K$ (we will use this at the very end of the proof). In the proof we use the short hand $X \times _ K Y$ for the fibre product over $\mathop{\mathrm{Spec}}(K)$. We will prove the lemma by induction on $\dim (X) + \dim (Y)$. The lemma is trivial if $\dim (X) \leq 0$, since in this case $U$ is an open and closed subscheme of $X$. Choose a point $z \in X \times _ K Y$. We will show the stalk at $\overline{z}$ is an isomorphism.

Suppose that $z \mapsto x \in X$ and assume $\text{trdeg}_ K(\kappa (x)) > 0$. Set $X' = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^{sh})$ and denote $U' \subset X'$ the inverse image of $U$. Consider the base change

\[ \xymatrix{ Y \times _ K X' \ar[d]_{q'} & Y \times _ K U' \ar[l]^{h'} \ar[d]^{p'} \\ X' & U' \ar[l]_{j'} } \]

of our diagram by $X' \to X$. Observe that $X' \to X$ is a filtered colimit of étale morphisms. By smooth base change in the form of Lemma 54.83.3 the pullback of $q^{-1}Rj_*\mathcal{F} \to Rh_*p^{-1}\mathcal{F}$ to $X'$ to $Y \times _ K X'$ is the map $(q')^{-1}Rj'_*\mathcal{F}' \to Rj'_*(p')^{-1}\mathcal{F}'$ where $\mathcal{F}'$ is the pullback of $\mathcal{F}$ to $U'$. (In this step it would suffice to use étale base change which is an essentially trivial result.) So it suffices to show that $(q')^{-1}Rj'_*\mathcal{F}' \to Rj'_*(p')^{-1}\mathcal{F}'$ is an isomorphism in order to prove that our original map is an isomorphism on stalks at $\overline{z}$. By Lemma 54.87.5 there is a separably closed field $L/K$ such that $X' = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ affine of finite type over $L$ and $\dim (X_ i) < \dim (X)$. For $i$ large enough there exists an open $U_ i \subset X_ i$ restricting to $U'$ in $X'$. We may apply the induction hypothesis to the diagram

\[ \vcenter { \xymatrix{ Y \times _ K X_ i \ar[d]_{q_ i} & Y \times _ K U_ i \ar[l]^{h_ i} \ar[d]^{p_ i} \\ X_ i & U_ i \ar[l]_{j_ i} } } \quad \text{equal to}\quad \vcenter { \xymatrix{ Y_ L \times _ L X_ i \ar[d]_{q_ i} & Y_ L \times _ L U_ i \ar[l]^{h_ i} \ar[d]^{p_ i} \\ X_ i & U_ i \ar[l]_{j_ i} } } \]

over the field $L$ and the pullback of $\mathcal{F}$ to these diagrams. By Lemma 54.80.3 we conclude that the map $(q')^{-1}Rj'_*\mathcal{F}' \to Rj'_*(p')^{-1}\mathcal{F}$ is an isomorphism.

Suppose that $z \mapsto y \in Y$ and assume $\text{trdeg}_ K(\kappa (y)) > 0$. Let $Y' = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^{sh})$. By Lemma 54.87.5 there is a separably closed field $L/K$ such that $Y' = \mathop{\mathrm{lim}}\nolimits Y_ i$ with $Y_ i$ affine of finite type over $L$ and $\dim (Y_ i) < \dim (Y)$. In particular $Y'$ is a scheme over $L$. Denote with a subscript $L$ the base change from schemes over $K$ to schemes over $L$. Consider the commutative diagrams

\[ \vcenter { \xymatrix{ Y' \times _ K X \ar[d]_ f & Y' \times _ K U \ar[l]^{h'} \ar[d]^{f'} \\ Y \times _ K X \ar[d]_ q & Y \times _ K U \ar[l]^ h \ar[d]^ p \\ X & U \ar[l]_ j } } \quad \text{and}\quad \vcenter { \xymatrix{ Y' \times _ L X_ L \ar[d]_{q'} & Y' \times _ L U_ L \ar[l]^{h'} \ar[d]^{p'} \\ X_ L \ar[d] & U_ L \ar[l]^{j_ L} \ar[d] \\ X & U \ar[l]_ j } } \]

and observe the top and bottom rows are the same on the left and the right. By smooth base change we see that $f^{-1}Rh_*p^{-1}\mathcal{F} = Rh'_*(f')^{-1}p^{-1}\mathcal{F}$ (similarly to the previous paragraph). By smooth base change for $\mathop{\mathrm{Spec}}(L) \to \mathop{\mathrm{Spec}}(K)$ (Lemma 54.84.1) we see that $Rj_{L, *}\mathcal{F}_ L$ is the pullback of $Rj_*\mathcal{F}$ to $X_ L$. Combining these two observations, we conclude that it suffices to prove the base change map for the upper square in the diagram on the right is an isomorphism in order to prove that our original map is an isomorphism on stalks at $\overline{z}$^{1}. Then using that $Y' = \mathop{\mathrm{lim}}\nolimits Y_ i$ and argueing exactly as in the previous paragraph we see that the induction hypothesis forces our map over $Y' \times _ K X$ to be an isomorphism.

Thus any counter example with $\dim (X) + \dim (Y)$ minimal would only have nonisomorphisms $q^{-1}Rj_*\mathcal{F} \to Rh_*p^{-1}\mathcal{F}$ on stalks at closed points of $X \times _ K Y$ (because a point $z$ of $X \times _ K Y$ is a closed point if and only if both the image of $z$ in $X$ and in $Y$ are closed). Since it is enough to prove the isomorphism locally, we may assume $X$ and $Y$ are affine. However, then we can choose an open dense immersion $Y \to Y'$ with $Y'$ projective. (Choose a closed immersion $Y \to \mathbf{A}^ n_ K$ and let $Y'$ be the scheme theoretic closure of $Y$ in $\mathbf{P}^ n_ K$.) Then $\dim (Y') = \dim (Y)$ and hence we get a “minimal” counter example with $Y$ projective over $K$. In the next paragraph we show that this can't happen.

Consider a diagram as in the statement of the lemma such that $q^{-1}Rj_*\mathcal{F} \to Rh_*p^{-1}\mathcal{F}$ is an isomorphism at all non-closed points of $X \times _ K Y$ and such that $Y$ is projective. The restriction of the map to $(X \times _ K Y)_{K^{sep}}$ is the corresponding map for the diagram of the lemma base changed to $K^{sep}$. Thus we may and do assume $K$ is separably algebraically closed. Choose a distinguished triangle

\[ q^{-1}Rj_*\mathcal{F} \to Rh_*p^{-1}\mathcal{F} \to Q \to (q^{-1}Rj_*\mathcal{F})[1] \]

in $D((X \times _ K Y)_{\acute{e}tale})$. Since $Q$ is supported in closed points we see that it suffices to prove $H^ i(X \times _ K Y, Q) = 0$ for all $i$, see Lemma 54.89.4. Thus it suffices to prove that $q^{-1}Rj_*\mathcal{F} \to Rh_*p^{-1}\mathcal{F}$ induces an isomorphism on cohomology. Recall that $\mathcal{F}$ is annihilated by $n$ invertible in $K$. By the Künneth formula of Lemma 54.89.2 we have

\begin{align*} R\Gamma (X \times _ K Y, q^{-1}Rj_*\mathcal{F}) & = R\Gamma (X, Rj_*\mathcal{F}) \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} R\Gamma (Y, \mathbf{Z}/n\mathbf{Z}) \\ & = R\Gamma (U, \mathcal{F}) \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} R\Gamma (Y, \mathbf{Z}/n\mathbf{Z}) \end{align*}

and

\[ R\Gamma (X \times _ K Y, Rh_*p^{-1}\mathcal{F}) = R\Gamma (U \times _ K Y, p^{-1}\mathcal{F}) = R\Gamma (U, \mathcal{F}) \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} R\Gamma (Y, \mathbf{Z}/n\mathbf{Z}) \]

This finishes the proof.
$\square$

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