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 a fieldan 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?
