The Stacks project

Proof. Let $Y \to X$ be a finite étale morphism. Consider the Stein factorization

of $Y \to S$. By Lemma 58.15.1 the morphism $T \to S$ is finite étale. In this way we obtain a functor $\textit{FÉt}_ X \to \textit{FÉt}_ S$. For any finite étale morphism $U \to S$ a morphism $Y \to U \times _ S X$ over $X$ is the same thing as a morphism $Y \to U$ over $S$ and such a morphism factors uniquely through the Stein factorization, i.e., corresponds to a unique morphism $T \to U$ (by the construction of the Stein factorization as a relative normalization in More on Morphisms, Lemma 37.53.1 and factorization by Morphisms, Lemma 29.53.4). Thus we see that the functors $\textit{FÉt}_ X \to \textit{FÉt}_ S$ and $\textit{FÉt}_ S \to \textit{FÉt}_ X$ are adjoints. Note that the Stein factorization of $U \times _ S X \to S$ is $U$, because the fibres of $U \times _ S X \to U$ are geometrically connected.

By the discussion above and Categories, Lemma 4.24.4 we conclude that $\textit{FÉt}_ S \to \textit{FÉt}_ X$ is fully faithful, i.e., $\pi _1(X) \to \pi _1(S)$ is surjective (Lemma 58.4.1).

It is immediate that the composition $\textit{FÉt}_ S \to \textit{FÉt}_ X \to \textit{FÉt}_{X_{\overline{s}}}$ sends any $U$ to a disjoint union of copies of $X_{\overline{s}}$. Hence $\pi _1(X_{\overline{s}}) \to \pi _1(X) \to \pi _1(S)$ is trivial by Lemma 58.4.2.

Let $Y \to X$ be a finite étale morphism with $Y$ connected such that $Y \times _ X X_{\overline{s}}$ contains a connected component $Z$ isomorphic to $X_{\overline{s}}$. Consider the Stein factorization $T$ as above. Let $\overline{t} \in T_{\overline{s}}$ be the point corresponding to the fibre $Z$. Observe that $T$ is connected (as the image of a connected scheme) and by the surjectivity above $T \times _ S X$ is connected. Now consider the factorization

Let $\overline{x} \in X_{\overline{s}}$ be any closed point. Note that $\kappa (\overline{t}) = \kappa (\overline{s}) = \kappa (\overline{x})$ is an algebraically closed field. Then the fibre of $\pi $ over $(\overline{t}, \overline{x})$ consists of a unique point, namely the unique point $\overline{z} \in Z$ corresponding to $\overline{x} \in X_{\overline{s}}$ via the isomorphism $Z \to X_{\overline{s}}$. We conclude that the finite étale morphism $\pi $ has degree $1$ in a neighbourhood of $(\overline{t}, \overline{x})$. Since $T \times _ S X$ is connected it has degree $1$ everywhere and we find that $Y \cong T \times _ S X$. Thus $Y \times _ X X_{\overline{s}}$ splits completely. Combining all of the above we see that Lemmas 58.4.3 and 58.4.5 both apply and the proof is complete. $\square$