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.
3.1. The connectedness theorem
The baby version of Zariski’s main theorem is a connectedness theorem about the fibers of a certain type of morphism.
Theorem 4 Let be a proper morphism of noetherian schemes. Suppose that the map is an isomorphism. Then the fibers of are connected.
In order to do this, we will apply the formal function theorem in the simplest case: when and ! Namely, then we have that
Now the first term is just the completion by the assumptions. Consequently, it is a complete local ring. The second term is also an -module, but if is disconnected into pieces , then it splits into two nontrivial pieces (the first piece generated by the section on and on , and the reversal for the second).
But a local ring is never decomposable as a module over itself. This would imply that there were proper ideals that summed to the entire ring. And this is a contradiction.
Corollary 5 (Baby Zariski) Let be a proper birational morphism between noetherian schemes. Suppose is normal. Then the fibers are connected.
Indeed, the claim is that in this case, . But is a finite -module and both have the same quotient field at each point, so normality implies that we have equality. Now the above theorem gives the result.
As before, let be a proper morphism of noetherian schemes, but let’s not assume anything more. We are going to define a factorization of into the composite of a map which always has connected fibers, and a finite morphism. To do this, note that is a finite -algebra, i.e. finite as a module, thanks to the direct image theorem. By the universal property of the relative , there is a factorization of :
where the second morphism is finite. Consider now the morphism . It is easy to check that since the push-forward to are the same and is finite. In particular, the fibers of are connected.
Theorem 6 (Stein factorization) Hypotheses as above, there is a factorization where is finite and has connected fibers. Further (where is the domain of ) is isomorphic to , and is isomorphic to .
We can now get a good picture of the connected components of the fiber of , only assuming is proper and all schemes are noetherian.
Corollary 7 The connected components of are in bijection with the maximal ideals of .
Indeed, with the same notation as above for the Stein factorization, we know that the connected components of are in bijection with the points in the fiber , because the fibers of are connected, and those of are discrete. But the points of lying above under are in bijection with the maximal ideals of In fact:
Corollary 8 Let be a dominant proper morphism of integral noetherian schemes. Then is naturally a subfield of . If , the number of connected components of is at most the number of maximal ideals of the integral closure of in .
As a result of this corollary, we can in fact strengthen the baby form of Zariski’s main theorem. In fact, let us say that a local domain is unibranch if its integral closure is also local. We can say that a point of a scheme is unibranch if its local ring is. Then, the above corollary implies that if is a birational proper morphism of noetherian schemes, for any unibranch , the fiber is connected.
So let’s prove this corollary. As before, we saw that the connected components of the fiber are in bijection with the maximal ideals of . However, this is a finite -module because is coherent. In particular, it is contained in the integral closure of in . Now every maximal ideal in the ring lifts to a different maximal ideal in by the lying over theorem. Thus, the result is clear.