This is the fifth (and last) post in a series on the fundamental group of a smooth curve in characteristic . It has been (for a blog) a somewhat long journey, so here again is the plan.
- One wishes to compute for a smooth proper curve over an algebraically closed field of characteristic . (For convenience, the geometric point defining it will sometimes be dropped.)
- One starts by lifting to characteristic zero, i.e. choosing a smooth proper scheme , for a complete DVR of unequal characteristic, whose special fiber is .
- One shows that and the special fiber have the same fundamental group; this uses formal GAGA to show that covers of can be lifted to covers of .
- 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 .
- 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 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 behaves nicely with respect to products, at least when one factor is proper. We first, however, need a result that states that calculating the of a variety in characteristic zero is equivalent to calculating it over . To prove this special case, we shall “bootstrap” by proving when both are varieties over (i.e. admit -points). Then we shall prove this (by a projective limit argument) when 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 be a proper, integral variety over an algebraically closed field . Let be a noetherian scheme. Let be a geometric point, and let be the composites to . Suppose is itself isomorphic to and the various maps are maps over . Then
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
which is proper and flat, with geometrically reduced fibers. One deduces from the Künneth formula that . As a result, we get a short exact sequence
The first term is , though, because the geometric fiber of under is just .
Now the result is “formal.” The natural map given above has a section (namely, the projection). It follows that there is a split exact sequence
with the first map admitting a section. It follows from this that the natural map must be an isomorphism. For, the above diagram shows it is injective, and surjectivity follows because we can construct a section by the inclusions using the geometric points as above.
Let us make an important remark here: the only place we used that was to argue that the geometric fiber of had the same fundamental group as that of . In general, the geometric fiber would be a base-change of 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 .
It follows now that if are finite type over an algebraically closed field and connected, and is proper over , then the above result holds for any geometric point , because the fundamental group is independent of the geometric point chosen, and because we can always find geometric points 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 be an inductive system of rings and a projective system of schemes over the projective system . Suppose the maps are of finite presentation, and are such that whenever , the diagram
is cartesian. Let be a geometric point and let be the images. Then the natural map
is an isomorphism.
Proof: This follows from the fact that to give an étale cover of is the same as giving a compatible family of étale covers of for large enough, which in turn is a general consequence of the “projective limit” formalism in EGA IV-8.
Lemma 23 Let be a proper, integral variety over an algebraically closed field . Let be an extension of which is also algebraically closed. Then, for any geometric point , the natural map
is an isomorphism.
Proof: Indeed, for each finitely generated integral -algebra contained in , we have that
We now take the projective limit as ranges over all finitely generated -algebras contained in . Then the latter term in the product tends to , 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 be a proper, integral variety over an algebraically closed field . Let be a noetherian scheme. Let be a geometric point, and let be the composites to . Then
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 admits topological generators.
We first claim:
Lemma 25 Consider a smooth projective curve where is algebraically closed of characteristic zero. Then admits topological generators.
Proof: Indeed, by “noetherian descent,” we note that is the colimit of the subfields which are the algebraic closures of finitely generated extensions of . As a result, there is a cartesian diagram for some :
Here, by choosing appropriately, noetherian descent allows us to assume that is itself a smooth projective curve. We know that the fundamental group of is the same as that of , so we can reduce to proving the result for . But embeds in , so we can base-change to . 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 be a smooth projective curve over a field of characteristic . Choose a smooth, proper lifting for a complete DVR of unequal characteristic. We know that (where we have omitted the geometric points for simplicity), so we are reduced to showing that is generated appropriately.
Then is flat and has geometrically reduced fibers (as a smooth morphism). Moreover, . To see this, note that is a finitely generated -module, and it is finite étale in view of the general fact about Stein factorizations; it is thus a product of copies of , which is complete. Since is integral, as a smooth scheme over a DVR, it follows that .
Let be a geometric point of mapping to the generic point. It follows that the exact sequence applies and the map
is surjective, because has no nontrivial étale covers. We have already seen that has topological generators; indeed, note that the genus of is that of , because the genus is constant in flat families by the semicontinuity theorem. It follows that has topological generators.