Recall that Zariski's Main Theorem states that if $f: X \to Y$ is a quasi-finite, separated, and finitely presented morphism into a quasi-compact separated scheme $Y$, then there is a factorization of $f$ into an open immersion followed by a finite morphism. In EGA IV-8, this is proved by reducing to the case of $Y$ the $\mathrm{Spec}$ of a noetherian ring by a finite presentation argument (the general machinery of which is developed in the prior part of that section), then reducing to the case of a local noetherian excellent ring (by again using the finite presentation argument, since by this machinery proving things about the local scheme $\mathrm{Spec}(\mathcal{O}_y)$ is the same as proving things in a neighborhood), and finally by completing and proving the result for $Y$ the spectrum of a complete local noetherian ring, after which it is basically commutative algebra.
This argument is very pretty, but I am curious if there is a more elementary approach in the special case of $Y$ noetherian, or even in the classical case of schemes of finite type over a field (that avoids the general machinery of finite presentation arguments and the descent of properties of morphisms under faithfully flat base-change). Namely, I am curious whether there is an argument that uses less fancy machinery, and could be phrased in the language of varieties. Is there one?
Best Answer
I would suggest chapter IV of the 1970 book "Anneaux Locaux Henséliens", by Michel Raynaud published in Springer Lecture Notes in Math no. 169. It gives a very general proof, way simpler than the one in EGA IV and, in my opinion, very readable. The proof is based in a paper by Peskine from 1966. The proof in Raynaud's book is complete, as far as I can recall.
As a footnote, sometimes noetherian hypothesis do not make arguments simpler, but, of course, this depends on the issue at hand.