Lemma 59.87.6. Let f : X \to S be a flat morphism of schemes such that for every geometric point \overline{x} of X the map
\mathcal{O}_{S, f(\overline{x})}^{sh} \longrightarrow \mathcal{O}_{X, \overline{x}}^{sh}
has geometrically connected fibres. Then for every cartesian diagram of schemes
\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }
with g quasi-compact and quasi-separated we have f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F} for any sheaf \mathcal{F} of sets on T_{\acute{e}tale}.
Proof.
It suffices to check equality on stalks, see Theorem 59.29.10. By Theorem 59.53.1 we have
(h_*e^{-1}\mathcal{F})_{\overline{x}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \times _ X Y, e^{-1}\mathcal{F})
and we have similarly
(f^{-1}g_*^{-1}\mathcal{F})_{\overline{x}} = (g_*^{-1}\mathcal{F})_{f(\overline{x})} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) \times _ S T, \mathcal{F})
These sets are equal by an application of Lemma 59.39.3 to the morphism
\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \times _ X Y \longrightarrow \mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) \times _ S T
which is a base change of \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) because Y = X \times _ S T.
\square
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