The existence of 2g generators for the fundamental group of a smooth curve

This is the fifth (and last) post in a series on the fundamental group of a smooth curve in characteristic $p$ . It has been (for a blog) a somewhat long journey, so here again is the plan.

One wishes to compute $\pi_1(X_0)$ for $X_0$ a smooth proper curve over an algebraically closed field of characteristic $p$ . (For convenience, the geometric point defining it will sometimes be dropped.)
One starts by lifting $X_0$ to characteristic zero, i.e. choosing a smooth proper scheme $X \to \mathrm{Spec } A$ , for $A$ a complete DVR of unequal characteristic, whose special fiber is $X_0$ .
One shows that $X$ and the special fiber have the same fundamental group; this uses formal GAGA to show that covers of $X_0$ can be lifted to covers of $X$ .
One develops an exact sequence intended to resemble the long exact sequence for a fibration in topology, which will enable us to obtain a relation between the fundamental groups of generic geometric fiber and that of $X$ .
In this post, we’ll show that the generic geometric fiber’s fundamental group is what we expect; this is basically because once you’re over characteristic zero, you can just base change to $\mathbb{C}$ without messing anything up. Yet this will take a bit of work to prove formally.
Finally, it’ll be straightforward to deduce the result!

1. The fundamental group of a product

We now want to show that étale ${\pi_1}$ behaves nicely with respect to products, at least when one factor is proper. We first, however, need a result that states that calculating the ${\pi_1}$ of a variety in characteristic zero is equivalent to calculating it over ${\mathbb{C}}$ . To prove this special case, we shall “bootstrap” by proving ${\pi_1(X \times_k Y) = \pi_1(X) \times \pi_1(Y)}$ when both are varieties over ${k}$ (i.e. admit ${k}$ -points). Then we shall prove this (by a projective limit argument) when ${Y}$ is the spectrum of a field, which is the result that changing the algebraically closed base field has no effect on the fundamental group (at least for a proper scheme).

Lemma 21 Let ${X \rightarrow \mathrm{Spec} k}$ be a proper, integral variety over an algebraically closed field ${k}$ . Let ${Y \rightarrow \mathrm{Spec} k}$ be a noetherian scheme. Let ${\overline{z} \rightarrow X \times_k Y}$ be a geometric point, and let ${\overline{x} \rightarrow X, \overline{y} \rightarrow Y}$ be the composites to ${X, Y}$ . Suppose ${\overline{z}}$ is itself isomorphic to ${\mathrm{Spec} k}$ and the various maps are maps over ${k}$ . Then

$\displaystyle \pi_1(X \times_k Y, \overline{z}) \simeq \pi_1(X, \overline{x}) \times \pi_1(Y, \overline{y})$

under the natural projection maps.

In the theory of the topological fundamental group, this result is immediate, but it is harder for the étale fundamental group.

Proof: We consider the map

$\displaystyle \pi: X \times_k Y \rightarrow Y,$

which is proper and flat, with geometrically reduced fibers. One deduces from the Künneth formula that ${\pi_*(\mathcal{O}_{X \times_k Y}) = \mathcal{O}_Y}$ . As a result, we get a short exact sequence

$\displaystyle \pi_1(X \times_k \overline{y} , \overline{y}) \rightarrow \pi_1(X \times_k Y, \overline{z}) \rightarrow \pi_1(Y, \overline{y}) \rightarrow 1.$

The first term is ${\pi_1(X, \overline{x})}$ , though, because the geometric fiber of ${X \times_k Y \rightarrow Y}$ under ${\mathrm{Spec} k = \overline{y} \rightarrow Y}$ is just ${X}$ .

Now the result is “formal.” The natural map given above ${\pi_1(X, \overline{x}) \rightarrow \pi_1(X \times_k Y, \overline{z})}$ has a section (namely, the projection). It follows that there is a split exact sequence

$\displaystyle 1 \rightarrow \pi_1(X , \overline{x}) \rightarrow \pi_1(X \times_k Y, \overline{z}) \rightarrow \pi_1(Y,\overline{y}) \rightarrow 1,$

with the first map admitting a section. It follows from this that the natural map ${\pi_1(X \times_k Y, \overline{z}) \rightarrow \pi_1(X, \overline{x}) \times \pi_1(Y, \overline{y})}$ must be an isomorphism. For, the above diagram shows it is injective, and surjectivity follows because we can construct a section by the inclusions ${X \rightarrow X \times_k Y, Y \rightarrow X \times_k Y}$ using the geometric points as above.

Let us make an important remark here: the only place we used that ${\overline{z} = \mathrm{Spec} k}$ was to argue that the geometric fiber of ${X \times_k Y}$ had the same fundamental group as that of ${X}$ . In general, the geometric fiber would be a base-change of ${X}$ to some other algebraically closed field. If we show that the algebraically closed base is irrelevant, then the above lemma will be seen to be true without any hypothesis on ${\overline{z}}$ .

It follows now that if ${X, Y}$ are finite type over an algebraically closed field ${k}$ and connected, and ${X}$ is proper over ${\mathrm{Spec} k}$ , then the above result holds for any geometric point ${\overline{z} \rightarrow X \times_k Y}$ , because the fundamental group is independent of the geometric point chosen, and because we can always find geometric points ${\mathrm{Spec} k \rightarrow X, Y}$ as they are “varieties.”

We now want to apply the general formalism of EGA IV-8 of “projective limits of schemes” to the étale fundamental group.

Lemma 22 Let ${\left\{A_\alpha\right\}}$ be an inductive system of rings and ${\left\{X_\alpha\right\}}$ a projective system of schemes over the projective system ${\left\{\mathrm{Spec} A_\alpha\right\}}$ . Suppose the maps ${X_\alpha \rightarrow \mathrm{Spec} A_\alpha}$ are of finite presentation, and are such that whenever ${\alpha \leq \beta}$ , the diagram

is cartesian. Let ${\overline{x} \rightarrow X}$ be a geometric point and let ${\overline{x}_\alpha \rightarrow X_\alpha}$ be the images. Then the natural map

$\displaystyle \pi_1(X, \overline{x}) \rightarrow \varprojlim \pi_1(X_\alpha, \overline{x}_\alpha)$

is an isomorphism.

Proof: This follows from the fact that to give an étale cover of ${X}$ is the same as giving a compatible family of étale covers of ${X_\alpha}$ for ${\alpha}$ large enough, which in turn is a general consequence of the “projective limit” formalism in EGA IV-8.

Lemma 23 Let ${X \rightarrow \mathrm{Spec} k}$ be a proper, integral variety over an algebraically closed field ${k}$ . Let ${K}$ be an extension of ${k}$ which is also algebraically closed. Then, for any geometric point ${\overline{x} \rightarrow X_K = X \times_k K}$ , the natural map

$\displaystyle \pi_1(X_K, \overline{x}) \rightarrow \pi_1(X, \overline{x})$

is an isomorphism.

Proof: Indeed, for each finitely generated integral ${k}$ -algebra contained in ${K}$ , we have that

$\displaystyle \pi_1(X \times_k \mathrm{Spec} A, \overline{x}) = \pi_1(X, \overline{x}) \times \pi_1(\mathrm{Spec} A, \overline{x}).$

We now take the projective limit as ${A}$ ranges over all finitely generated ${k}$ -algebras contained in ${K}$ . Then the latter term in the product tends to ${\pi_1(\mathrm{Spec} K, \overline{x}) =1}$ , so we get the result.

As indicated in the end of the proof, this allows us to generalize the product formula by removing the hypothesis on the geometric point.

Corollary 24 Let ${X \rightarrow \mathrm{Spec} k}$ be a proper, integral variety over an algebraically closed field ${k}$ . Let ${Y \rightarrow \mathrm{Spec} k}$ be a noetherian scheme. Let ${\overline{z} \rightarrow X \times_k Y}$ be a geometric point, and let ${\overline{x} \rightarrow X, \overline{y} \rightarrow Y}$ be the composites to ${X, Y}$ . Then

$\displaystyle \pi_1(X \times_k Y, \overline{z}) \simeq \pi_1(X, \overline{x}) \times \pi_1(Y, \overline{y})$

under the natural projection maps.

2. Completion of the proof

Finally, we have all the main ingredients necessary to complete the proof that the étale fundamental group of a smooth projective curve of genus ${g}$ admits ${2g}$ topological generators.

We first claim:

Lemma 25 Consider a smooth projective curve ${X_0 \rightarrow \mathrm{Spec} k}$ where ${k}$ is algebraically closed of characteristic zero. Then ${\pi_1(X_0, \overline{x})}$ admits ${2g}$ topological generators.

Proof: Indeed, by “noetherian descent,” we note that ${k}$ is the colimit of the subfields ${k' \subset k}$ which are the algebraic closures of finitely generated extensions of ${\mathbb{Q}}$ . As a result, there is a cartesian diagram for some ${k'}$ :

Here, by choosing ${k'}$ appropriately, noetherian descent allows us to assume that ${X_0' \rightarrow \mathrm{Spec} k'}$ is itself a smooth projective curve. We know that the fundamental group of ${X_0}$ is the same as that of ${X_0'}$ , so we can reduce to proving the result for ${X_0'}$ . But ${k' }$ embeds in ${\mathbb{C}}$ , so we can base-change to ${\mathrm{Spec} \mathbb{C}}$ . Then, however, the result is a consequence of the Riemann existence theorem and the ordinary theory of the topological fundamental group for a compact topological surface.

It is now clear how to finish the proof of Grothendieck’s theorem, following the “scheme” sketched above. Namely, let ${X_0 \rightarrow \mathrm{Spec} k}$ be a smooth projective curve over a field of characteristic ${p}$ . Choose a smooth, proper lifting ${X \rightarrow \mathrm{Spec} A}$ for ${A}$ a complete DVR of unequal characteristic. We know that ${\pi_1(X) = \pi_1(X_0)}$ (where we have omitted the geometric points for simplicity), so we are reduced to showing that ${\pi_1(X)}$ is generated appropriately.

Then ${X \rightarrow \mathrm{Spec} A}$ is flat and has geometrically reduced fibers (as a smooth morphism). Moreover, ${f_*(\mathcal{O}_X) = \mathcal{O}_{\mathrm{Spec} A}}$ . To see this, note that ${\Gamma(X, \mathcal{O}_X) = B}$ is a finitely generated ${A}$ -module, and it is finite étale in view of the general fact about Stein factorizations; it is thus a product of copies of ${A}$ , which is complete. Since ${X}$ is integral, as a smooth scheme over a DVR, it follows that ${\Gamma(X, \mathcal{O}_X) = A}$ .

Let ${\overline{\xi}}$ be a geometric point of ${\mathrm{Spec} A}$ mapping to the generic point. It follows that the exact sequence applies and the map

$\displaystyle \pi_1(X_{\overline{\xi}}) \rightarrow \pi_1(X)$

is surjective, because ${\mathrm{Spec} A}$ has no nontrivial étale covers. We have already seen that ${\pi_1(X_{\overline{\xi}})}$ has ${2g }$ topological generators; indeed, note that the genus of ${X_{\overline{\xi}}}$ is that of ${X_{0}}$ , because the genus is constant in flat families by the semicontinuity theorem. It follows that ${\pi_1(X) = \pi_1(X_0)}$ has ${2g}$ topological generators.