# How to get the connectedness theorem from the quasi-finite version of ZMT?

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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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.)