Lemma 59.86.5. Consider a cartesian diagram of schemes

where $g : T \to S$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be an abelian sheaf on $T_{\acute{e}tale}$. Let $q \geq 0$. The following are equivalent

For every geometric point $\overline{x}$ of $X$ with image $\overline{s} = f(\overline{x})$ we have

\[ H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S T, \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S T, \mathcal{F}) \]$f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is an isomorphism.

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