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 commute

Spec(K) ---> X | | | | f v v Spec(R) ---> Ythere 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