Lemma 59.39.4. Let $S$ be a scheme. Let $f : X \to S$ be a morphism such that
$f$ is submersive, and
the geometric fibres of $f$ are connected.
Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. Then $\Gamma (S, \mathcal{F}) = \Gamma (X, f^{-1}_{small}\mathcal{F})$.
Proof. There is a canonical map $\Gamma (S, \mathcal{F}) \to \Gamma (X, f_{small}^{-1}\mathcal{F})$. Since $f$ is surjective (because its fibres are connected) we see that this map is injective.
To show that the map is surjective, let $\tau \in \Gamma (X, f_{small}^{-1}\mathcal{F})$. It suffices to find an étale covering $\{ U_ i \to S\} $ and sections $\sigma _ i \in \mathcal{F}(U_ i)$ such that $\sigma _ i$ pulls back to $\tau |_{X \times _ S U_ i}$. Namely, the injectivity shown above guarantees that $\sigma _ i$ and $\sigma _ j$ restrict to the same section of $\mathcal{F}$ over $U_ i \times _ S U_ j$. Thus we obtain a unique section $\sigma \in \mathcal{F}(S)$ which restricts to $\sigma _ i$ over $U_ i$. Then the pullback of $\sigma $ to $X$ is $\tau $ because this is true locally.
Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s}$ in $S$. Consider the image of $\tau $ in the stalk
See Lemma 59.36.2. We can find an étale neighbourhood $U \to S$ of $\overline{s}$ and a section $\sigma \in \mathcal{F}(U)$ mapping to this image in the stalk. Thus after replacing $S$ by $U$ and $X$ by $X \times _ S U$ we may assume there exits a section $\sigma $ of $\mathcal{F}$ over $S$ whose image in $(f_{small}^{-1}\mathcal{F})_{\overline{x}}$ is the same as $\tau $.
By Lemma 59.39.1 there exists a maximal open $W \subset X$ such that $f_{small}^{-1}\sigma $ and $\tau $ agree over $W$ and the formation of $W$ commutes with further pullback. Observe that the pullback of $\mathcal{F}$ to the geometric fibre $X_{\overline{s}}$ is the pullback of $\mathcal{F}_{\overline{s}}$ viewed as a sheaf on $\overline{s}$ by $X_{\overline{s}} \to \overline{s}$. Hence we see that $\tau $ and $\sigma $ give sections of the constant sheaf with value $\mathcal{F}_{\overline{s}}$ on $X_{\overline{s}}$ which agree in one point. Since $X_{\overline{s}}$ is connected by assumption, we conclude that $W$ contains $X_ s$. The same argument for different geometric fibres shows that $W$ contains every fibre it meets. Since $f$ is submersive, we conclude that $W$ is the inverse image of an open neighbourhood of $s$ in $S$. This finishes the proof. $\square$
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