Lemma 59.87.1. Consider the cartesian diagram of schemes

$\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

Assume that $f$ is flat and every object $U$ of $X_{\acute{e}tale}$ has a covering $\{ U_ i \to U\}$ such that $U_ i \to S$ factors as $U_ i \to V_ i \to S$ with $V_ i \to S$ étale and $U_ i \to V_ i$ quasi-compact with geometrically connected fibres. Then for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.

Proof. Let $U \to X$ be an étale morphism such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale and $U \to V$ quasi-compact with geometrically connected fibres. Observe that $U \to V$ is flat (More on Flatness, Lemma 38.2.3). We claim that

\begin{align*} f^{-1}g_*\mathcal{F}(U) & = g_*\mathcal{F}(V) \\ & = \mathcal{F}(V \times _ S T) \\ & = e^{-1}\mathcal{F}(U \times _ X Y) \\ & = h_*e^{-1}\mathcal{F}(U) \end{align*}

Namely, thinking of $U$ as an object of $X_{\acute{e}tale}$ and $V$ as an object of $S_{\acute{e}tale}$ we see that the first equality follows from Lemma 59.39.31. Thinking of $V \times _ S T$ as an object of $T_{\acute{e}tale}$ the second equality follows from the definition of $g_*$. Observe that $U \times _ X Y = U \times _ S T$ (because $Y = X \times _ S T$) and hence $U \times _ X Y \to V \times _ S T$ has geometrically connected fibres as a base change of $U \to V$. Thinking of $U \times _ X Y$ as an object of $Y_{\acute{e}tale}$, we see that the third equality follows from Lemma 59.39.3 as before. Finally, the fourth equality follows from the definition of $h_*$.

Since by assumption every object of $X_{\acute{e}tale}$ has an étale covering to which the argument of the previous paragraph applies we see that the lemma is true. $\square$

 Strictly speaking, we are also using that the restriction of $f^{-1}g_*\mathcal{F}$ to $U_{\acute{e}tale}$ is the pullback via $U \to V$ of the restriction of $g_*\mathcal{F}$ to $V_{\acute{e}tale}$. See Sites, Lemma 7.28.2.

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