The valuative criterion quantifies over all valuation rings

Assume the Axiom of Choice (The Axiom of Choice) and let $f:X\to S$ be quasi-separated. The criterion proved on this page tests separatedness by uniqueness of lifts of valuative diagrams (Valuative uniqueness diagram) whose ring is an arbitrary valuation ring (Valuation rings) $R$ with fraction field $K$, with no Noetherian, finite-type, finiteness or dimension restriction (Valuative uniqueness detects separatedness).

The quantifier cannot be narrowed to discrete valuation rings without extra hypotheses. A companion calculation on the examples page glues two copies of $\mathrm{Spec}V$ along $\mathrm{Spec}K$, where $V={\bigcup }_{n\ge 1}k[t^{1\mathrm{/}n}{]}_{(t^{1\mathrm{/}n})}$ inside $K={\bigcup }_{n\ge 1}k(t^{1\mathrm{/}n})$. The result is a quasi-separated morphism that is not separated over $\mathrm{Spec}V$, yet every valuative diagram over a discrete valuation ring (Discrete valuation rings) has at most one lift; the diagram over the rank-one valuation ring $V$, whose value group is $\mathbb{Q}$, has two lifts, one for each glued closed point. The mechanism is arithmetic: a local homomorphism $V\to A$ into a DVR $A$ whose generic map factors through $K$ sends $t$ to a nonzero element, since the field map $K\to \mathrm{Frac}(A)$ is injective. It would send every $t^{1\mathrm{/}n}$ into the maximal ideal, so the finite positive integer $v_A(t)=n{v}_A(t^{1\mathrm{/}n})$ would have to be divisible by every $n\ge 1$, which is impossible. Local maps that kill $t$, such as $V\to k\to k[[u]]$, do exist; their generic image is a closed point of the glued scheme and they do not give this obstruction. In that case both lifts land in the same chart and the fixed map to $\mathrm{Spec}V$ determines them uniquely.

A DVR-only criterion is nevertheless a theorem in the context that makes the reduction possible: for a morphism of finite type between locally Noetherian schemes, separatedness is equivalent to at-most-one lift for every diagram over a discrete valuation ring (Vakil, Theorem 13.7.1). No hypothesis of that kind is present in the general criterion, and none is available for an arbitrary quasi-separated morphism, so no such reduction may be used implicitly.

Finally, the general criterion also ranges over valuation rings that happen to be fields, and those diagrams never obstruct uniqueness: if $R=K$ is a field then $\mathrm{Spec}R=\mathrm{Spec}K$ is the source of the generic map, and the only compatible lift is that generic map itself. What the counterexample exploits is therefore not the exclusion of fields from Discrete valuation rings but the absence of any discreteness or finiteness hypothesis on the valuation ring; a restatement of the criterion that quantifies only over DVRs is false without the finite-type and locally Noetherian hypotheses above.