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
where is finite and has connected fibers. The fibers of are discrete, and . The connected component of in is thus . We thus find that is isolated in its fiber if and only if . So we may look at the corresponding set for ; it suffices to prove the theorem for .
That is, we may prove the proposition under the assumption that the fibers are all connected, and furthermore that the pushforward of the structure sheaf is the structure sheaf (since that is true of ). Let us now drop the notation , and assume that itself has connected fibers and satisfies .
We need to show that the set of such that is open. If there is such an , I claim that . This follows because as becomes a small neighborhood of , then becomes a small neighborhood of as is closed. Thus, since , we find that the natural map
is an isomorphism. However, this means that there are small neighborhoods of which are isomorphic under . By choosing small, we can just take . It follows then that every point in is isolated in its fiber. This proves the result.
In fact, we have shown more:
Proposition 10 Let be a proper morphism of noetherian schemes satisfying . Then if is the set of which are isolated in their fibers, is an open immersion.
Indeed, we have seen that as defined above is an open set in the proof of the previous proposition. We actually saw more, though, in the proof: if , then there are neighborhoods of and of , with , such that is an isomorphism. If we take the inverse , and do the same for every such pair that occurs around any , then the various inverses have to glue.
Finally, we can state the second version of the Zariski theorem.
Theorem 11 (Zariski) Let be a quasiprojective morphism of noetherian schemes. Let be the subset of consisting of points which are isolated in their fibers. Then factors as a composite of an open immersion and a finite morphism.
To see this, first note that by definition of quasiprojective, factors as a composite of an open immersion and a projective morphism. Consequently we can just assume is projective. In this case, we have the Stein factorization where is finite. But on the set of points which are isolated in their fibers, the above corollary shows that is actually an open immersion. So we find
is the requisite factorization. We can also prove:
Theorem 12 A proper, quasi-finite morphism of noetherian schemes is finite.
Now it turns out that a quasi-finite morphism is in fact quasi-projective, but this is very nontrivial. We are not yet in a position to prove this.
However, we can at least appeal to the Stein factorization theorem to get where is finite and is an open immersion, since every point is isolated in its fiber. However, is proper, and is separated, so by the cancellation property, we find that is itself proper. This means that it must be a closed immersion as well, and consequently finite. It follows that is finite.