The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it satisfies the following criterion:
For any valuation ring R (with K=Frac(R)) and any morphisms Spec(R)→Y and Spec(K)→X making the following square commuteSpec(K) ---> X | | | | f v v Spec(R) ---> Y
there exists at most one (resp. exactly one) morphism Spec(R)→X filling in the diagram.
But if Y is locally noetherian and f:X→Y is of finite type, then this condition only needs to be verified for discrete valuation rings. Does anybody know of an example where it is not sufficient to use DVRs? In other words, does there exist a morphism of schemes f:X→Y which is not separated (resp. proper), but does satisfy the valuative criterion for DVRs?
Reference: EGA II, Proposition 7.2.3 and Theorem 7.3.8