The second version of Zariski’s main theorem

There is another version of Zariski’s main theorem, intermediate between the baby one and the hard quasi-finite one. It basically is the big Zariski theorem but for quasi-finite and quasi-projective morphisms. The main point of the argument is a careful application of Stein factorization, discussed yesterday; I’d like to discuss that here. What one eventually has to see is that the quasi-projective hypothesis can be dropped, but this takes some work.

As before, let be a proper morphism of noetherian schemes. We are now interested in the question of when some is isolated in the fiber .

Proposition 9 If is proper, then the set of isolated in their fiber is open.

To see this, we shall use the Stein factorization. As before, we can write

 

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.