The Stacks project

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

\[ \pi _1(X_{\overline{s}}) \to \pi _1(X) \to \pi _1(S) \to 1 \]

of fundamental groups. See Proposition 58.15.2.

reference

Lemma 58.15.1. 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 37.52.5. If $f$ is of finite presentation, flat, with geometrically reduced fibres, then $S' \to S$ is finite étale.

Proof. This follows from Derived Categories of Schemes, Lemma 36.32.8 and the information contained in More on Morphisms, Theorem 37.52.5. $\square$

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

\[ \pi _1(X_{\overline{s}}) \to \pi _1(X) \to \pi _1(S) \to 1 \]

of fundamental groups.

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

\[ \xymatrix{ Y \ar[d] \ar[r] & X \ar[d] \\ T \ar[r] & S } \]

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 37.52.1 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

\[ \pi : Y \longrightarrow T \times _ S X \]

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. $\square$


Comments (2)

Comment #2807 by BL on

Silly typo, in third line from the end: "and we find tat" -- "tat" should be "that."


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