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Suppose f:X→Y is a morphism of finite type between locally noetherian schemes. The valuative criterion for separatedness (resp. properness) says roughly that f is a separated (resp. proper) morphism if and only if the following condition holds:

For any curve C in Y and for any lift of C-{p} to X, there is at most one (resp. exactly one) way to extend this to a lift of C to X.

More precisely,

If C is the spectrum of a DVR with closed point p (a very local version of a curve: the intersection of all open neighborhoods of p on an "honest" curve), C→Y is a morphism, and C-{p}→X is a lift of that morphism along f, there is at most one (resp. exactly one) way to complete it to a lift C→X.

Does it suffice to check the valuative criteria on an even more local kind of object: the spectrum of a complete DVR? This would be quite nice because the only complete DVRs over a field k are rings of the form L[[t]] where L is an extension of k.

More generally, if you drop the hypotheses that f is of finite type and X and Y are locally noetherian, the usual valuative criteria must be verified for arbitrary valuation rings. Is it enough to check them for complete valuation rings?

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"...the only complete DVRs are rings of the form L[[t]]". Hmm. I think another way of saying that is "I am not a number theorist" :-) –  Kevin Buzzard Nov 8 '09 at 8:08
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@buzzard: corrected. But please don't be so coy. If you see something wrong, say so, rather than just making a joke about it. –  Anton Geraschenko Nov 8 '09 at 14:49
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1 Answer

up vote 4 down vote accepted

Yep, a quasi-compact morphism of schemes (resp. locally noetherian schemes) is universally closed if and only the existence part of the valuative criterion holds for complete valuation rings (resp. complete DVRs) with algebraically closed residue field.

This is in EGA (see II.7.3.8 and the remark II.7.3.9). Note that the separated hypothesis is not necessary there; for the valuative criterion of properness one needs to require that the morphism is quasi-separated.

This holds more generally for Artin stacks if one allows a field extension of the fraction field of the valuation ring (see LMB 7.3).

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One has to be a bit careful with the notion "complete valuation rings" for non-discrete valuations. Complete here does not mean m-adic'ally complete (with "complete" := "separated and complete"). In fact, if I remember correctly, the m-adic completion of a non-noetherian valuation ring is always zero. –  David Rydh Oct 5 '09 at 0:08
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