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…)

Here are a bunch of notes I wrote up over winter break. The notes are intended to cover Grothendieck’s argument for Zariski’s Main Theorem (the quasi-finite version). It contains as subsections various blog posts I’ve done here, but also a fair bit of additional material. For instance, they cover some of the basic properties of unramified and étale morphisms of rings. It turns out that to prove things about them, you need ZMT in some form.  I wrote some of this up here too.

As written, the notes are still incomplete. Many arguments (such as the use of fpqc descent and the filtered colimit argument) are currently sketched. Someday I may expand these notes to be more complete; right now I don’t think I have the time. Still, I think the basic outline of what happens is present.

On MathOverflow, Kevin Buzzard famously remarked that as a graduate student he was confused by the numerous forms of Hilbert’s Theorem 90, thinking at one point that it was a practical joke: the result was what one would invoke whenever one was stuck.

I actually feel the same way about Zariski’s Main Theorem in algebraic geometry. Having made a couple of unsuccessful attempts by now at reading Mumford’s book on abelian varieties, I was struck at how often this seemingly ubiquitous result was invoked repeatedly. Later on MathOverflow, I learned from BCnrd that ZMT is the “engine” behind proving things about certain properties of schemes: for instance, proving that locally étale morphisms have a given form. So I need to understand this result.

What it states is succint: a quasi-finite morphism of separated noetherian schemes factors as a composite of an open immersion and a finite morphism. This is a fairly big deal, as the condition of quasi-finiteness is seemingly rather weak—it’s a condition on the fibers—while open immersions and finite morphisms are very nice. While it is easy to state, the general form of ZMT (due to Grothendieck) is fairly difficult; it is in EGA IV-3. I am going to try starting with the “baby” version of Zariski’s main theorem (in EGA III-4 or Hartshorne), which runs as follows.

Theorem 3 (Zariski) Let ${f: X \rightarrow Y}$ be a birational projective morphism of noetherian integral schemes, where ${Y}$ is normal. Then the fibers ${f^{-1}(y) = X_y, y \in Y}$ are all connected.

This is tricky. A priori, we know that for any open subset ${U \subset Y}$, the inverse image ${f^{-1}(U) \subset X}$ is open and thus connected as ${X}$ is irreducible. As the ${U}$‘s shrink towards ${y \in Y}$, we might expect the “limit” of the ${f^{-1}(U)}$ to be connected. However, this doesn’t work. The ${U}$‘s that contain ${y}$ are actually rather large, since we are working with the Zariski topology. The problem is that Zariski neighborhoods are rather large, and so, intuitively, one might think to consider completions. In fact, this is what we are going to do: we will deduce the result from the formal function theorem. (more…)