As of late, I’ve been reading the proof due to Laumon of the Weil conjectures (a simplification of Deligne’s second proof) via the Fourier-Deligne transform. This is quite interesting, and I’d like to start a series of posts on it soon, based on the notes I’ve been taking. I don’t currently have the time to edit the notes, so I’ll devote this post to a curious fact I learned today.

One of the first results one proves when studying the classical fundamental group in topology is that the fundamental group of a topological group is abelian. As I learned today, the analogous result for the etale fundamental group fails.

Let $G_0$ be a smooth group scheme, defined over a finite field $\kappa$ of size $q$, and let $G$ be the base-change to the algebraic closure. Then there is the Lang map $L: G \to G$ sending $x \mapsto (Fx) x^{-1}$, for $F$ the Frobenius.

It is a theorem (of Lang) that this map $L$ is a surjective map. It is also finite, since the fiber over the identity is finite (the $\kappa$-rational points), and a morphism of homogeneous spaces for an algebraic group with finite fibers is finite. (Quick but probably unnecessarily non-elementary proof: a morphism of reduced homogeneous spaces over a smooth group scheme is faithfully flat, by generic flatness and a translation argument. As a result, it’s a quotient map. Since the fibers are finite, one can check that the map is closed by a similar translation argument. Then, proper plus finite fibers implies finite by Zariski’s Main Theorem.)

Also, the Frobenius induces the zero map on the tangent spaces. As a result, $x \mapsto (Fx) x^{-1}$, the Lang morphism, is smooth as the morphism on tangent spaces is a bijection. So, since the fibers are finite, we have an etale cover!

The claim is that it is a Galois cover, and the automorphism group is $G_0(\kappa)$: indeed, the right translations by the $\kappa$-rational points are automorphisms of the cover. Since this is the right number of translations for the degree of the cover, we have indeed a Galois cover with the appropriate Galois group. But, in general, this Galois group won’t be abelian. So the fundamental group, which surjects onto every Galois group, can’t be abelian either.

I initially tried to prove the false result by the Eckmann-Hilton argument (until I was told about this counterexample), but it seems not to be correct: perhaps the problem is that the etale fundamental group doesn’t commute with products! (It does for proper schemes over an algebraically closed field, but this is very nontrivial.)

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! (more…)

This is the fourth in a series of posts started here intended to describe Grothendieck’s work on the fundamental group of smooth curves in positive characteristic. Past posts were devoted to showing that a smooth proper curve $X_0$ could always be “lifted” to characteristic zero, via a proper smooth map $X \to \mathrm{Spec} A$ (with $A$ a complete discrete valuation ring) whose specific fiber is the initial curve $X_0$. The promise was to use various methods of algebraic geometry to compare the fundamental groups of the “generic” and the “special” fibers. In the previous post, we handled the case of the special fiber, where we argued that the fundamental group of the special fiber was the same as that of the whole thing. In this post, we want to develop the technology to handle the general fiber.

Our next goal is to obtain a small analog of the classical long exact sequence of a fibration in homotopy theory. Namely, the smooth proper morphism ${X \rightarrow \mathrm{Spec} A}$ constructed as earlier that lifts the curve ${X_0 \rightarrow \mathrm{Spec} k}$ will be the “fibration,” and we are going to take the (geometric) fiber over the generic point. Since the base ${\mathrm{Spec} A}$ has trivial ${\pi_1}$ (because ${A}$ is complete local and its residue field is algebraically closed), it will follow from this long exact sequence that

$\displaystyle \pi_1(X_{\overline{\xi}}) \rightarrow \pi_1(X)$

is a surjection. (more…)

(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.

One of the early achievements of Grothendieck’s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective curve in characteristic ${p}$. The result is that if ${X_0}$ is a curve of genus ${g}$ over an algebraically closed field of characteristic ${p}$, then its fundamental group is topologically generated by ${2g}$ generators. This is analogous to the characteristic zero case, where the topological fundamental group generated by ${2g}$ generators (subject to a single relation) by the theory of surfaces. Since this result uses many interesting techniques, I thought I would devote a few posts to discussing it.

To motivate étale ${\pi_1}$, let’s recall the following statement:

Theorem 1 Let ${X}$ be a nice topological space, and ${x_0 \in X}$. Then the functor ${p^{-1}(x_0)}$ establishes an equivalence of categories between covering spaces ${p: \overline{X} \rightarrow X}$ and ${\pi_1(X, x_0)}$-sets.

This is one way of phrasing the Galois correspondence between subgroups of ${\pi_1(X, x_0)}$ and connected covering spaces of ${X}$, but which happens to be more categorical and generalizable. The interpretation of ${\pi_1(X, x_0)}$ as classifying covering spaces is ultimately the one that will work in an algebraic context. One can’t talk about homotopy classes of loops in an algebraic variety. However, Grothendieck showed:

Theorem 2 Let ${X}$ be a connected scheme, and ${\overline{x_0} \rightarrow X}$ a geometric point. Then there is a unique profinite group ${\pi_1(X, \overline{x_0})}$ such that the fiber functor of liftings ${\overline{x_0} \rightarrow \overline{X}}$ establishes an equivalence of categories between (finite) étale covers ${p: \overline{X} \rightarrow X}$ and finite continuous ${\pi_1(X, x_0)}$-sets.