Valuative uniqueness detects separatedness

Since $f$ is quasi-separated, [F5] makes $\mathrm{\Delta }$ quasi-compact, and [F4] makes it an immersion.

If $f$ is separated, [F3] gives the uniqueness property for every valuative diagram; it remains to prove the converse.

Assume now that every valuative diagram for $f$ has at most one lift, and suppose for contradiction that $\mathrm{\Delta }(X)$ is not closed in $P$. By [F7], applied to the quasi-compact immersion $\mathrm{\Delta }$ of step 1.1 under the Axiom of Choice, there are $z\in \mathrm{\Delta }(X)$ and $t\in \overline{\mathrm{\Delta }(X)}\setminus \mathrm{\Delta }(X)$ with $t\in \overline{\{z\}}$.

By [F11] there is a unique $x_0\in X$ with $z=\mathrm{\Delta }(x_0)$, and by the residue clause of [F4] the two projections induce one and the same isomorphism $\kappa (x_0)\to \kappa (z)$, whose inverse is induced by $\mathrm{\Delta }$; so we may regard $K:=\kappa (z)$ as identified with $\kappa (x_0)$.

Because $t\in \overline{\{z\}}$, every open neighbourhood of $t$ contains $z$; choose an affine open $W=\mathrm{Spec}R$ containing $t$, so $z\in W$. Writing $t\leftrightarrow \mathfrak{p}$ and $z\leftrightarrow \mathfrak{q}$, the relation $t\in \overline{\{z\}}$ is exactly $\mathfrak{q}\subseteq \mathfrak{p}$. By [F10] the local ring ${\mathcal{O}}_{P,t}$ is $R_{\mathfrak{p}}$ and $K=\kappa (z)=\mathrm{Frac}(R\mathrm{/}\mathfrak{q})$; the composite $R_{\mathfrak{p}}\to R_{\mathfrak{p}}\mathrm{/}\mathfrak{q}{R}_{\mathfrak{p}}\subseteq \mathrm{Frac}(R\mathrm{/}\mathfrak{q})=K$ is a local ring map, its image $A=(R\mathrm{/}\mathfrak{q}{)}_{\mathfrak{p}\mathrm{/}\mathfrak{q}}$ is a local subring of $K$, and the maximal ideal of $A$ is the image of ${\mathfrak{m}}_t=\mathfrak{p}{R}_{\mathfrak{p}}$.

By [F8] there is a valuation ring $V\subseteq K$ with fraction field $K$ dominating $A$ in the sense of [F12]; composing the local map ${\mathcal{O}}_{P,t}\to A\subseteq V$ with the inclusion gives a local homomorphism ${\mathcal{O}}_{P,t}\to V$, which by [F9] corresponds to a morphism $\mathrm{Spec}V\to P$.

Under that morphism the generic point of $\mathrm{Spec}V$ maps to $z$ and the closed point maps to $t$: by [F9] the morphism $\mathrm{Spec}V\to P$ built in step 6.1 corresponds to the pair consisting of the point $t$ and the local homomorphism ${\mathcal{O}}_{P,t}\to V$, so its closed point is $t$ and the induced residue-field map is the canonical one $\kappa (t)\to V\mathrm{/}{\mathfrak{m}}_V$, which is injective because both sides are fields; the generic point is the image of the field-valued point $\mathrm{Spec}K\to \mathrm{Spec}V\to P$, whose local map ${\mathcal{O}}_{P,t}\to V\to K$ is the map of step 5.1 with kernel $\mathfrak{q}{R}_{\mathfrak{p}}$, so by [F9] and [F10] it is the point $z$ with residue field $\kappa (z)=K$.

Let $a,b:\mathrm{Spec}V\to X$ be the composites of $\mathrm{Spec}V\to P$ with ${\mathrm{pr}}_1,{\mathrm{pr}}_2$, and let $i:\mathrm{Spec}K\to \mathrm{Spec}V$ be the canonical morphism. By step 7.1 the generic point of $\mathrm{Spec}V$ maps to $z$, so $a\circ i$ and $b\circ i$ correspond to the two maps $\kappa (x_0)\to \kappa (z)=K$, which coincide by step 4.1; hence $a\circ i=b\circ i$. Also $f\circ a=f\circ b$, since both are the structure map $\mathrm{Spec}V\to P\to S$. Thus the data form a valuative diagram for $f$ with generic map $a\circ i=b\circ i$ and two lifts $a,b$.

The two lifts are distinct: at the closed point of $\mathrm{Spec}V$ the maps $a,b$ take the values ${\mathrm{pr}}_1(t)$ respectively ${\mathrm{pr}}_2(t)$, and their residue-field maps to $V\mathrm{/}{\mathfrak{m}}_V$ factor through the two maps into $\kappa (t)$ induced by the projections and the injective field map $\kappa (t)\to V\mathrm{/}{\mathfrak{m}}_V$. Since $t\mathrm{\notin }\mathrm{\Delta }(X)$ the clause of [F4] fails for $t$: either ${\mathrm{pr}}_1(t)\mathrm{\ne }{\mathrm{pr}}_2(t)$ as points of $X$, in which case $a,b$ differ at the closed point, or these points are equal to a point $x$ and the two induced maps of residue fields $\kappa (x)\to \kappa (t)$ differ, in which case composing with the injective map $\kappa (t)\to V\mathrm{/}{\mathfrak{m}}_V$ shows that $a,b$ induce different residue-field maps at the closed point; in both cases the morphisms $a,b$ differ. This contradicts the assumed uniqueness for the diagram of step 8.1.

Therefore $\mathrm{\Delta }(X)$ is closed in $P$. Since $\mathrm{\Delta }$ is an immersion by step 1.1, [F6] makes $\mathrm{\Delta }$ a closed immersion, and then [F2] says that $f$ is separated.

Steps 2.1 and 10.1 prove the equivalence under AC; the Axiom of Choice was used exactly in the boundary specialization [F7] and the dominating valuation ring [F8], no existence of lifts was asserted, and the quantification over all valuation rings includes fields and non-Noetherian rank-one rings.