(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.

I don’t really anticipate doing all that much serious blogging for the next few weeks, but I might do a few posts like this one.

First, I learned from Qiaochu in a comment that the commutativity of the endomorphism monoid of the unital object in a monoidal category can be proved using the Eckmann-Hilton argument. Let $1$ be this object; then we can define two operations on $End(1)$ as follows.  The first is the tensor product: given $a,b$, define $a.b := \phi^{-1} \circ a \otimes b \circ \phi$, where $\phi: 1 \to 1 \otimes 1$ is the isomorphism.  Next, define $a \ast b := a \circ b$.  It follows that $(a \ast b) . (c \ast d) = (a . c) \ast (b. d)$ by the axioms for a monoidal category (in particular, the ones about the unital object), so the Eckmann-Hilton argument that these two operations are the same and commutative. (more…)