When can the "homotopy exact sequence" of etale fundamental groups for a smooth curve fail to be exact?

Question

Suppose you have a smooth proper curve $f : \overline{X}\rightarrow S$ over an arbitrary locally noetherian scheme $S$ with a section $e : S\rightarrow \overline{X}$. Let $X := \overline{X} - e(S)$. Let $s\in S$ be a geometric point, and $x\in X_s$ also a geometric point.

Suppose all residue characteristics of $S$ are 0 (so $S$ is a scheme over $\mathbb{Q}$), then must the following sequence (of etale fundamental groups) be exact? $$1\rightarrow \pi_1(X_s,x)\rightarrow\pi_1(X,x)\rightarrow\pi_1(S,s)\rightarrow 1$$ This is proven to be true in SGA I, Expose XIII (Exemples 4.4) when $X\rightarrow S$ admits a section $g : S\rightarrow X$, though the proof seems quite technical so before I invest time into seriously analyzing it, I'd appreciate if anyone can give some examples for where things can go wrong.

More generally, let $L$ be the set of primes which do not appear as residue characteristics of $S$, let $\pi_1^L(X_s,x)$ be the pro-$L$ completion of $\pi_1(X_s,x)$, and let $\pi_1'(X,x)$ be the maximal quotient of $\pi_1(X,x)$ such that the kernel of $\pi_1'(X,x)\rightarrow \pi_1(S,s)$ is pro-$L$.

When can the sequence $$1\rightarrow \pi_1^L(X_s,x)\rightarrow\pi_1'(X,x)\rightarrow\pi_1(S,s)\rightarrow 1$$ fail to be exact?

Answer 1

I will address injectivity of $\pi_1(X_s) \to \pi_1(X)$ in the characteristic zero case. Exactness at the other places needs a proof too, but is easier (and you can find it in SGA).

I assume that the genus of $\overline{X}_s$ is $g > 0$ as the case $g = 0$ is trivial.

Let $S' \to S$ be a finite \'etale morphism with $S'$ connected. Let $X' = X \times_S S'$. Then $X'$ is connected too (by the arguments used to prove the exactness in the other places). Then $\pi_1(X') \to \pi_1(X)$ is injective, hence it suffices to prove $\pi_1(X_s) \to \pi_1(X')$ is injective.

Pick an integer $n > 2$. After a replacement as in the previous paragraph we may assume the monodromy action on $H^1_{\acute{e}tale}(\overline{X}_s, \mathbf{Z}/n\mathbf{Z})$ is trivial. We may also assume that $S$ is a scheme over $\mathbf{Q}(\zeta_n)$. Thus there is a morphism $S \to {}_n\mathcal{M}_{g, 1}$ such that $\overline{X}$ and its section are the pullback from the universal family. Here ${}_n\mathcal{M}_{g, 1}$ is the moduli stack of $1$-pointed smooth projective genus $g$ curves endowed with a "symplectic" basis of $n$-torsion on its Jacobian. By our choice of $n$ the algebraic stack ${}_n\mathcal{M}_{g, 1}$ is a smooth scheme ${}_nM_{g, 1}$. By functoriality of the fundamental group we reduce to the case $S = {}_nM_{g, 1}$ over $\mathbf{Q}(\zeta_n)$.

You can look at Petersen's answer above and insert your favorite computation of $\pi_2({}_nM_{g, 1})$ here but I want to give a more elementary proof.

In the universal case the question can be translated into a question about moduli of curves. I will do this for you without giving you all the details, but I think you will be convinced. Namely, let $G$ be a finite group. Consider the moduli stack ${}_{G, n}\mathcal{M}_{g, 1}$ parametrizing connected finite \'etale $G$-covers $D \to C$ with $C = \overline{C} \setminus \{c\}$ where $\overline{C}$ is a smooth projective genus $g$ curve endowed with a "symplectic" basis of $n$-torsion in its Jacobian. The natural map $$ {}_{G, n}\mathcal{M}_{g, 1} \longrightarrow {}_n\mathcal{M}_{g, 1} $$ is \'etale by deformation theory and proper by the valuative criterion along dvrs (it is not finite bc it may not be representable). To finish the proof we just need to show that ${}_{G, n}\mathcal{M}_{g, 1}$ is a stacky quotient of the form $[M'/H]$ where $M' \to {}_nM_{g, 1}$ is a finite \'etale morphism of schemes and $H$ is a finite group acting on $M'$ over ${}_nM_{g, 1}$. This is clear because we can let $M'$ be the moduli scheme parametrizing some rigidification of the problem defining ${}_{G, n}\mathcal{M}_{g, 1}$, for example add in a "symplectic" basis for the $n$-torsion on the Jacobian of $D$.

Why does this prove the result? Well, it shows that any $G$-cover of the geometric generic fibre of the universal curve spreads out over some finite \'etale cover $M'$ of the moduli space. Cheers!

Answer 2

I expect the existence of such a short exact sequence to fail already classically for the usual fundamental group. Let's work over the complex numbers. If $X \to S$ has a section then $\pi_2(X) \to \pi_2(S)$ is surjective and then from the homotopy fibration sequence $$ \pi_2(X) \to \pi_2(S) \to \pi_1(X_s) \to \pi_1(X) \to \pi_1(S) \to \pi_0(X_s) = 1$$ you get the sequence you're after. But in general I see no reason to believe $\pi_1(X_s) \to \pi_1(X)$ injective.

Answer 3

Let me make some comments on how the question plays out when we look at topology of finite-type schemes over $\mathbb{C}$. In this case, the usual comparison theorem provides an isomorphism $$ \widehat{\pi_1^{top}(X(\mathbb{C}))}\cong \pi_1^{et}(X). $$

First, an explicit standard example complementing Dan Petersen's answer: the Hopf fibration $\mathbb{C}^2\setminus\{0\}\to\mathbb{CP}^1$ with fiber $\mathbb{C}^\times$ gives an example where the map $\pi_2(\mathbb{CP}^1)\to\pi_1(\mathbb{C}^\times)$ is in fact an isomorphism, so that there is no exact sequence of fundamental groups.

Second, let me note that the Hopf fibration example actually does not really fit the special situation outlined at the beginning of the question. Over $\mathbb{C}$ one can show exactness of the sequence of topological fundamental groups for smooth finite-type base as follows. If $\overline{X}\to S$ is a smooth proper family of curves over a complex manifold $S$, it is in fact topologically locally trivial by Ehresmann, i.e., there exists a topological covering $\tilde{S}\to S$ such that the family becomes trivial. Now restrict to the open family $X=\overline{X}\setminus e(S)\to S$. After base change along $\tilde{S}\to S$ we have $X\times_S\tilde{S}\cong C\times \tilde{S}$ and then a surjection (for curves actually an isomorphism) $\pi_2(X\times_S\tilde{S})\to\pi_2(\tilde{S})$. Since covering projections induce isomorphisms on homotopy groups in degrees $n\geq 2$, we know that the natural map $\pi_2(X)\to\pi_2(S)$ is also surjective (for curves an isomorphism). Hence we get an exact sequence of topological fundamental groups.

Of course, this does not yet answer the question. First, there are issues with profinite completion not necessarily being exact (as mentioned in the comment of Elden Elmanto). This involves both suitable finiteness assumptions as well as nilpotency conditions for the $\pi_1(S)$-action on the cohomology of the fiber. Further issues are with the smoothness assumption in the application of Ehresmann's fibration theorem. And, of course, this argument has the problem that it's not directly clear how to algebraize.