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.

  1. 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.)
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.

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