The baby version of Zariski’s main theorem

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 be a birational projective morphism of noetherian integral schemes, where is normal. Then the fibers are all connected.

This is tricky. A priori, we know that for any open subset , the inverse image is open and thus connected as is irreducible. As the ‘s shrink towards , we might expect the “limit” of the to be connected. However, this doesn’t work. The ‘s that contain 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.

 

Formal functions II

So last time, we introduced the first form of the formal function theorem. We said that if was a proper scheme over with structure morphism , and for some ideal , then there were two constructions one could do on a coherent sheaf on that were in fact the same. Namely, we could complete the cohomology with respect to , and we could take the inverse limit . The claim was that the natural map

was in fact an isomorphism. This is a very nontrivial statement, but in fact we saw yesterday that a reasonably straightforward proof could be given via diagram-chasing if one appeals to a strong form of the proper mapping theorem.

1. Formal functions, jazzed up

Now, however, we want to jazz this up a little. I won’t do this as much as possible because I don’t want to talk too much about formal schemes yet. On the other hand, I want to replace cohomology groups with higher direct images.

 

The formal function theorem

(Well, it looks like I should stop making promises on this blog. There hasn’t been a single post about spectra yet. I hope that will change before next semester.)

So, today I am going to talk about the formal function theorem. This is more or less a statement that the properties of taking completions and taking cohomologies are isomorphic for proper schemes. As we will see, it is the basic ingredient in the proof of the baby form of Zariski’s main theorem. In fact, this is a very important point: the formal function theorem allows one to make a comparison with the cohomology of a given sheaf over the entire space and its cohomology over an “infinitesimal neighborhood” of a given closed subset. Now localization always commutes with cohomology on non-pathological schemes. However, taking such “infinitesimal neighborhoods” is generally too fine a job for localization. This is why the formal function theorem is such a big deal.

I will give the argument following EGA III here, which is more general than that of Hartshorne (who only handles the case of a projective scheme). The form that I will state today is actually rather plain and down-to-earth. In fact, one can jazz it up a little by introducing formal schemes; perhaps this is worth discussion next time.