# A noetherian proof of Zariski's Main Theorem?

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?

• Have you tried looking for Zariski's original proof? (I assume it was Zariski who originally proved it, anyway.) Dec 26, 2010 at 16:49
• I don't think Zariski proved the quasifinite version (which I believe is due to Grothendieck); the original result was about birational correspondences where the target was normal. Dec 26, 2010 at 17:11
• Look in Mumford's Red Book. Dec 26, 2010 at 18:13

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.

There's a purely algebraic proof in some lecture notes by Mel Hochster. He explains the translation into the language of varieties, as well.

• Thanks! Can the quasi-finite version of ZMT be deduced from the algebraic one? Dec 26, 2010 at 16:31
• There is a second reformulation of the result in those notes that probably resembles the statement you're looking for a bit more. Dec 26, 2010 at 16:44
• @Harry: I'm pretty sure that the second version you're referring to can be deduced directly from Stein factorization (it's Cor. 4.4.6 in EGA III, except that the finite algebra is not proved to be a subring, and Grothendieck is in the noetherian case). Namely the morphism $\mathrm{Spec} S \to \mathrm{Spec} R$ is finite type, and the prime $Q$ is isolated in its fiber, so (by Stein factorization thm.) locally the map is an open immersion followed by a finite morphism. I believe this is weaker than the general ZMT. Dec 26, 2010 at 17:18
• @Harry: I mean, the key thing that isn't at all obvious is that a quasi-finite morphism is quasiprojective. If it's quasiprojective, then formal function arguments are enough. Dec 26, 2010 at 17:27
• (Also, it's probably possible to make the finite algebra a subring just by quotienting out by the appropriate ideal. E.g. wlog assume that the Stein factorization is as a dominant open immersion followed by a finite morphism.) Dec 26, 2010 at 17:30

Raynaud and Hochster, and Stacks, give essentially the proof of Peskine this proof does not use noetherianity

a constructive proof, extracted from Peskine proof is given in the following paper

{Alonso, M. E. and Coquand, T. and Lombardi, H.}, TITLE = {Revisiting {Z}ariski main theorem from a constructive point of view}, FJOURNAL = {Journal of Algebra}, VOLUME = {406}, YEAR = {2014}, PAGES = {46--68},