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Let $f: X \to Y$ be a proper morphism of noetherian schemes. If the natural map $\mathcal{O}_Y \to f_*(\mathcal{O}_X)$ is an isomorphism, then a version of Zariski's main theorem states that the fibers $X_y, y \in Y$ are all connected. (The case listed as "Zariski's main theorem" in Hartshorne is the case of $f$ birational and $Y$ normal.) This may be proved via the formal function theorem.

The fancier version of ZMT (EGA IV-8) is that a quasifinite separated morphism of finite presentation between quasicompact, quasiseparated schemes factors as the composite of an open immersion and a finite morphism.

Is there a direct way to deduce the connectedness theorem from the more general ZMT?

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up vote 4 down vote accepted

I'm not sure that this is directly possible. If memory serves, Zariski's original version of his main theorem showed something like the following: if $f: X \to Y$ is birational with $Y$ normal, and if $y \in Y$ is a point where $f^{-1}$ is not defined (I think this is what Zariski calls a fundamental point), then each component of $f^{-1}(y)$ is positive dimensional. This result can be proved using Grothendieck's form of ZMT, I think: if $f^{-1}(y)$ contains an isolated point, we can choose a n.h. of this point such that the restriction of $f$ to this n.h. has finite fibres, hence is an isomorphism (here we are using Grothendieck's ZMT together with normality of $Y$), and so in fact $f^{-1}$ can be defined at $y$ after all. (This is quite possibly not quite correct, but hopefully is not totally bogus either, both in the argument and in the claim to some historical accuracy. Also, I think that if you look e.g. in the commutative algebra part of the stacks project a version of ZMT along these lines is discussed; at least, there are results in which the hypothesis of an isolated point in the fibre plays a prominent role.)

Zariski's connectedness theorem came quite a bit later, in his monograph on formal functions, and the techniques were quite a bit more involved. (They were a precursor to formal scheme techniques, in which one can complete in some directions but not in others, as opposed to earlier techniques with complete local rings, in which one completes in every direction around a point at once.)

A quick glance over Mumford's discussion of ZMT in the Red Book (which is always a good place to go to for learning ZMT intuition) suggests that I'm not blundering here. (Grothendieck's formulation of ZMT is what he calls version IV of ZMT, while the connectedness theorem is his version V, and he singles out version V as being "more global" than the other versions, and doesn't discuss any implications from the other versions to this one.)

[On the other hand, this contradicts the wikipedia entry, which suggests that Grothendieck's formulation does imply the connectedness theorem. (But doesn't quite say how.) There are plenty of people at Harvard you could ask for clarification on this point, of course ... .]

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Thanks for the answer! I believe your argument. (For some reason, I was under the impression that the connectedness theorem would just follow at once from the general ZMT, probably because I was under the impression that the latter was much harder.) – Akhil Mathew Dec 25 '10 at 15:56

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