(This post is deliberately inflammatory. I’ll make statements a little stronger than I normally would, in the hopes of encouraging a response.)

I’d like to make the controversial case that the proof of the valuative criteria (of separatedness and properness) can be profitably be removed from a first course, or at least a course such as this (meaning from any course that is not aimed at extremely well-prepared students). I’m open to it being available for people to read, and intend to do that in these notes.

Reasons:

(i) Students find the proofs harder than you might think (or remember) — the vast majority of students in such a class don’t have experience with valuation rings coming in.

(ii) They never need to be used in the class. In fact, whenever they are used, I claim that it is faster not to use them.
People may disagree with this, so some details are given below.

(iii) I agree that they give the right way to picture separated and proper morphisms. But that’s an argument for
stating them, not for proving them and using them to prove basic properties. (For comparison, I like
telling people that quasifinite morphisms in reasonable situations are always just open immersions into finite morphisms. But no one in their right mind would prove Zariski’s Main Theorem when quasifinite morphisms are introduced.)

(iv) I agree that the valuative criteria are useful, and are used by many algebraic geometers. But anyone is free to read the proofs later.

A little more detail on (ii): it’s true that valuative criteria give quick proofs of things, such as the fact that projective morphisms are proper. But the proof of the fundamental theorem of elimination theory (translation: that projective morphisms are closed) is shorter and low-tech — it is tricky, but certainly much easier than the valuative criterion. (If you’re going to state the valuative criterion, and not prove it, and use it to prove some basic properties, you may as well just state and not prove the fundamental theorem of elimination theory.) Elimination theory has the advantage of being clearly of prior interest to (most) students. I should also admit than when learning algebraic geometry,
I found the valuative criteria hard, and thus I thought separatedness and properness were difficult notions. I was surprised to find later (when reading EGA) that in fact both were very simple notions and easy to use, and that the
main thing in learning them is to come to terms with the fact that they are actually natural nice notions.

Note also that in these notes, discrete valuation rings (let alone general valuation rings) are not yet anywhere in sight, yet we can already deal happily and harmlessly with separatedness and properness (and will, in the next batch of notes).

(A little more on (iv): when do people first use — in a necessary way — valuative criteria? They certainly come up in moduli theory, but later than you might think. For example, the valuative criterion is not needed to show that the Hilbert scheme is projective — at most you use the “valuative criterion for a locally closed subscheme of P^n_Z to be closed”. The first place I think of them as being necessary is in figuring out the right definition of properness for a Deligne-Mumford stack. In other words, I think they could be dealt with after introducing stacks, which certainly shouldn’t be in a first course.)

So my question: can someone make a convincing argument in favor of the valuative criteria?
More generally: please argue with me!

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