Lemma 58.15.1.reference Let $f : X \to S$ be a proper morphism of schemes. Let $X \to S' \to S$ be the Stein factorization of $f$, see More on Morphisms, Theorem 37.53.5. If $f$ is of finite presentation, flat, with geometrically reduced fibres, then $S' \to S$ is finite étale.
58.15 Homotopy exact sequence
In this section we discuss the following result. Let $f : X \to S$ be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume $S$ is connected and let $\overline{s}$ be a geometric point of $S$. Then there is an exact sequence
of fundamental groups. See Proposition 58.15.2.
Proof. This follows from Derived Categories of Schemes, Lemma 36.32.8 and the information contained in More on Morphisms, Theorem 37.53.5. $\square$
Proposition 58.15.2. Let $f : X \to S$ be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume $S$ is connected and let $\overline{s}$ be a geometric point of $S$. Then there is an exact sequence
of fundamental groups.
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. $\square$
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