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Notes on Zariski’s Main Theorem, étaleness and unramifiedness, and constructible sets

Posted by Akhil Mathew under

algebraic geometry | Tags:

etale morphisms,

unramified morphisms,

Zariski's main theorem |

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Here are a bunch of notes I wrote up over winter break. The notes are intended to cover Grothendieck’s argument for Zariski’s Main Theorem (the quasi-finite version). It contains as subsections various blog posts I’ve done here, but also a fair bit of additional material. For instance, they cover some of the basic properties of unramified and étale morphisms of rings. It turns out that to prove things about them, you need ZMT in some form. I wrote some of this up here too.

As written, the notes are still incomplete. Many arguments (such as the use of fpqc descent and the filtered colimit argument) are currently sketched. Someday I may expand these notes to be more complete; right now I don’t think I have the time. Still, I think the basic outline of what happens is present.

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