(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!

December 4, 2010 at 5:32 pm

Sorry to disappoint you, but I whole-heartedly agree!

When I first saw the valuative-criteria proof that projective space is proper, that made almost no mark on me, because it offloaded the major work to some other theorem whose proof was difficult and not memorable. On the other hand, properness of projective space is such a basic, beautiful and concrete fact and I found it far more illuminating to see a bare-hands proof of it.

December 4, 2010 at 6:49 pm

I totally see what you are saying. But… here is a fun algebra fact which totally “explains” the valuative criterion (for u.c. of q.c. morphisms):

If A —> B is a ring map such that the image of Spec(B) in Spec(A) is a subset closed under specialization, then the image is closed. Proof: We may assume A —> B is injective. Then the map is dominant so all generic points are in the image. As the image is closed under specialization the image is closed.

Maybe the relationship between specialization and valuation rings is not so clear to students at first. But if you only do the minimum and define a valuation ring to be a subring R of a field K which is local (and not a field) and maximal for the relation of domination, then the existence result needed for the valuative criterion is a standard application of Zorn’s lemma.

If you want to avoid the valuative criterion now, then surely later you’ll want to say things like: a variety X is proper iff every rational map from a nonsingular curve C to X extends to a morphism C —> X. And when you prove this the notion of domination comes up naturally.

So maybe an alternative is to prove the valuative criterion in terms of specializations of points now and address how this is related to valuation rings later?

PS: In your proof of the fundamental theorem of elimination theory can you please explain more carefully how high a power of x_i you have to multiply with to get your a’_i and also what is the “graded ideal of functions vanishing on Z”. Also, why does the f you find do the job? (Although this should be pretty clear once you’ve explained what is I(Z)_. me thinks.)

December 5, 2010 at 6:48 pm

Dear Johan,

I like this approach via explicit discussion of specialization, with valuation rings and dominance naturally appearing later when one translates things back to algebraic language.

Best wishes,

Matt

December 4, 2010 at 8:33 pm

Ravi, there are two useful principles in the proof of the valuative criteria: (i) the situation of one point specializing to another is always dominated by a valuation ring accomplishing the same property (even a dvr in the loc. noetherian case), (ii) closedness of constructible sets can be detected using specialization of points. These are very useful for a student to have in mind when studying specialization later in life (i.e., the valuation ring or dvr cases are “no worse” than the general case). But these can be noted without getting into the details, which may be left to the industrious students to understand via homework reading.

I agree that students benefit from not overdoing the uses of the valuative criteria. However, these criteria should not be given short shrift, since they are an ideal early example of how one really does use the functorial viewpoint to establish properties of “abstract” spaces and morphisms. It’s great practice with functorial reasoning. To prove properness of projective space over Spec(Z), and not just in the case over a field, do you really have an elimination-theoretic proof? Anyway, of course the real point of valuative criteria is seen only in the study of moduli spaces, and serious moduli schemes do not arise in a first course in schemes, but there are plenty of examples of this way before stacks.

How do you prove properness of the Jacobian of a smooth proper family of geom. connd curves (or even a single smooth proper geom. connd curve over a field)? Or properness for Pic^0 in other situations? Or for Hilbert schemes independently of the construction (which is always a good thing to do, for better conceptual insight; I dislike proofs “by staring at the construction”)? Or for finiteness of the $j$-map from a modular curve to the affine line when working over Spec(Z[1/N])? And later in life, properness for Hilbert functors as algebraic spaces (logically comes prior to DM stacks)? The closedness of the Plucker map can be seen “by hand”, and that is both illuminating and useful (since the valuative criterion proof doesn’t apply in analytic case), but to give students practice with functorial reasoning why not have them *also* work out the valuative proof in homework, to see diverse points of view?

I strongly disagree with saying that these criteria are needed to find the right defn of properness in the setting of stacks: the definition has nothing at all to do with valuative criterion. Indeed, the defn is almost the same as the scheme case, via universal closedness and properness of the diagonal. Perhaps you meant to say that these criteria are needed to have the right intuition for how to think about separatedness of a moduli scheme/stack and how to search for a compactified moduli scheme/stack? I agree with that, but in the stack case (as for schemes) the sufficiency of valuative criteria is a serious theorem, not a definition (and anyone claim to take it as a definition is playing word games by shifting the real work to somewhere else).

To summarize: one of the important skills that students should get out of a year-long course on schemes is the ability to apply (and to appreciate!) functorial reasoning in the study of the structure of schemes. Valuative criteria are a very important example of this kind of insight, and one which they can get see in action that way pretty early on. So why not give them ample practice with it in this way (but not in the service of the elementary properties of separatedness and properness, which is a “bad” application)?

December 5, 2010 at 12:24 pm

A response from Brian Osserman, sent to me by email:

[begin quote]

– I have no problem with omitting the proof from lecture, if it is

available for the students to read. But in general, I have little problem

with omitting proofs. I could see arguments either way, in this case: you

and I find the proof intimidating because valuation rings are not as

ubiquitous these days. But they remain important for certain arguments,

and in many subfields, and one could argue that at least giving an

overview of the proof would serve as a worthwhile introduction to them.

– I agree that one should not apply the valuative criteria just for the

sake of applying it, especially at the cost of imposing unnecessary

Noetherian hypotheses.

– However, I do like the proof of properness of projective space (and for

that matter, Grassmannians) using the valuative criterion. I don’t claim

that it’s easier per se, but I find the elimination theory approach to

be unenlightening.

– To me, the comparison to quasifinite morphisms and Zariski’s main theorem

is specious. Zariski’s main theorem is way, way harder to prove, and I would

also argue that the definition of quasifinite is in fact more intuitive than

the definition of finite anyhow (i.e., I think of the theorem that finite

is equivalent to proper and quasifinite as clarifying finite morphisms).

– The valuative criterion is not necessary for defining properness of stacks,

and in fact neither Deligne and Mumford nor Laumon-Moret-Bailly define it

this way (DM’s definition seems slightly ad hoc, while LMB’s is a natural

generalization of the scheme definition).

What it comes down to for me ultimately is that the main importance of

the valuative criteria in an intro course is twofold: if presented properly

(a huge if), it can provide important intuition for what properness and

separatedness mean; also, it gets students thinking in directions that are

very useful when dealing with moduli spaces, and in particular serves as an

introduction to functorial criteria for properties of morphisms (as formal

smoothness, etc) without requiring talking about functors.

[end quote]

[begin quote]

– The valuative criterion is not necessary for defining properness of stacks,

and in fact neither Deligne and Mumford nor Laumon-Moret-Bailly define it

this way (DM’s definition seems slightly ad hoc, while LMB’s is a natural

generalization of the scheme definition).

Although it’s not terribly relevant to defining properness, perhaps what

you were getting at here is the following: even when dealing with moduli

spaces, as long as we stick with functors and are proving representability,

we can frequently get by by proving that a couple of them are proper and

constructing the others in terms of those. However, once we start working

with stacks this seems much less the case, so we find ourselves using the

valuative criteria more frequently.

[end quote]

December 11, 2010 at 5:34 am

(As usual, I am putting more here before even having the chance to properly digest earlier comments. But now that grades are about to be submitted, I am starting to empty the inbox faster than it is being filled, and should keep to my promise of gradually catching up starting in the new year.)

There is one point no one has made, because likely it is clear to them. One direction of the valuative criterion of separatedness is repeatedly useful, and will be repeatedly used. But that will be shown in an easy result that maps from a reduced scheme to a separated scheme are determined by their behavior on a dense open subset of the source.

December 19, 2010 at 4:54 pm

I didn’t have any appreciation of valuative criteria until seeing it connected to normality; “if a rational function doesn’t blow up in codim 1, then it should be a global function” = normality. (Vs. “if a rational function is well-defined in codim 1, it should be global” = S2.) But I don’t use it otherwise much, and basically agree with you.

January 8, 2011 at 8:20 am

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March 26, 2012 at 9:12 pm

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