June 19, 2011

Specializing the fundamental group I

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
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(This is the third post in a project started here to describe some of Grothendieck’s work on the fundamental group of curves in positive characteristic.)

Let {X_0 \rightarrow \mathrm{Spec} k} be a smooth curve over an algebraically closed field {k} of characteristic {p}. We are interested in determining a set of topological generators for this curve. To do this, we started by showing that if {A} is a complete DVR with residue field {k}, then one can “lift” (by using cohomological vanishing and formal-to-algebraic comparison theorems) {X_0 \rightarrow \mathrm{Spec} k} to a smooth, proper morphism {X \rightarrow \mathrm{Spec} A}.

Ideally, we will have chosen {A} to be characteristic zero itself. Now our plan is to compare the two geometric fibers of {X}: one is {X_0}, and the other is {X_{\overline{\xi}}} (where {\xi} is the generic point; the over-line indicates that one wishes an algebraic closure of {k(\xi) = K(A)} here) with each other. Ultimately, we are going to show two things:

 

  1. The natural map {\pi_1(X_0) \rightarrow \pi_1(X)} is an isomorphism.
  2. The natural map {\pi_1(X_{\overline{\xi}}) \rightarrow \pi_1(X)} is an epimorphism.

Here we have been loose with notation, as we have not indicated the relevant geometric points. The geometric point is, however, irrelevant for a connected scheme.

It will follow from this that there is an continuous epimorphism of profinite groups

\displaystyle \pi_1(X_{\overline{\xi}}) \rightarrow \pi_1(X_0).

However, {\pi_1(X_{\overline{\xi}})} will be seen to be topologically generated by {2g} generators (where {g} is the genus) by comparison with a curve over {\mathbb{C}}. For a smooth curve of genus {g} over {\mathbb{C}}, it is clear from the Riemann existence theorem (and the topological fundamental group) that {\pi_1} has {2g} topological generators.

Thus, it will follow, as stated earlier:

Theorem  If {X_0} is a smooth curve of genus {g} over an algebraically closed field of any characteristic, {\pi_1(X_0)} is topologically generated by {2g} generators.

 

One technical point will be, of course, that it is not entirely obvious that {\pi_1(X_{\overline{\xi}})} is the same as it would be for a curve over {\mathbb{C}}. This requires independent proof, but it will not be too hard.

(more…)

 

June 18, 2011

Lifting smooth curves to characteristic zero

Posted by Akhil Mathew under algebraic geometry | Tags: , , |
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This is a continuation of the project outlined in this post yesterday of describing Grothendieck’s proof that the fundamental group of a smooth curve in characteristic p has 2g topological generators (where g is the genus). The first step, as I explained there, is to show that one may “lift” such smooth curves to characteristic zero, in order that a comparison may be made between the characteristic p curve and something much more concrete in characteristic zero, that we can approach via topological methods. This post will be devoted to showing that such a lifting is always possible.

1. Introduction

It is a general question of when one can “lift” varieties in characteristic {p} to characteristic zero. Doing so often allows one to bring in transcendental techniques (to the lift), as it will in this case of {\pi_1}. Let us thus be formal:

Definition 4 Let {X_0} be a proper, smooth scheme of finite type over a field {k} of characteristic {p}. We say that a lifting of {X_0} is the data of a DVR {A} of characteristic zero with residue field {k}, and a proper, smooth morphism {X \rightarrow \mathrm{Spec} A} whose special fiber is isomorphic to {X_0}.

There are obstructions that can prevent one from making such a lifting. One example is given by étale cohomology. A combination of the so-called proper and smooth base change theorems implies that, in such a situation, the cohomology of the special fiber and the cohomology of the general fiber, with coefficients in any finite group without {p}-torsion, are isomorphic. As a result, if there is something funny in the étale cohomology of {X_0}, it might not be liftable. See this MO question.

In the case of curves, fortunately, it turns out there are no such problems, but still actually lifting one will take some work. We aim to prove:

Theorem 5 Let {X_0 \rightarrow \mathrm{Spec} k} be a smooth, proper curve of finite type over the field {k} of characteristic {p}. Then if {A} is any complete DVR of characteristic zero with residue field {k}, there is a smooth lifting {X \rightarrow \mathrm{Spec} A} of {X_0}.

One should, of course, actually check that such a complete DVR does exist. But this is a general piece of algebra, found for instance in Serre’s Local Fields.

The reason there won’t be any obstructions in the case of curves is that they are of dimension one, but we’ll see that the cohomological obstructions to lifting all live in {H^2}.

The strategy, in fact, will be to lift {X_0 \rightarrow \mathrm{Spec} k} to a sequence of smooth schemes {X_n \rightarrow \mathrm{Spec} A/\mathfrak{m}^n} (where {\mathfrak{m} \subset A} is the maximal ideal) that each lift each other, using the local nilpotent lifting property of smooth morphisms.

This family {\left\{X_n \rightarrow \mathrm{Spec} A/\mathfrak{m}^n\right\}} is an example of a so-called formal scheme, which for our purposes is just such a compatible sequence of liftings. Obviously any scheme {X \rightarrow \mathrm{Spec} A} gives rise to a formal scheme (take the base-changes to {\mathrm{Spec} A/\mathfrak{m}^n}), but it is actually nontrivial (i.e., not always true) to show that a formal scheme is indeed of this form. But we will be able to do this as well in the case of curves. (more…)