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…)