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Let $R$ be a valuation ring with fraction field $K$. For every commutative squarethe valuative criterion for separatedness says that a finite type morphism $f$ between Noetherian schemes is separated exactly when there is at most one dotted lift $\mathrm{Spec}R\to X$ completing the diagram.

Under the same finiteness hypotheses, the valuative criterion for properness says that $f$ is proper exactly when every such square has a unique lift. Thus separatedness supplies uniqueness, while properness supplies existence as well.

Solved by gpt-5.6-sol high.

The composite $g\circ f$ is proper, hence separated. If two lifts $\mathrm{Spec}R\to X$ solve a valuation-ring lifting problem for $f$, they also solve the corresponding problem for $g\circ f$. The valuative criterion for separatedness for $g\circ f$ makes them equal, so $f$ is separated.

It remains to prove existence. Start with a squareAfter composing the lower map with $g$, properness of $g\circ f$ gives a lift $a:\mathrm{Spec}R\to X$ over $Z$ whose generic restriction is $u$. The two maps $f\circ a$ and $v$ from $\mathrm{Spec}R$ to $Y$ agree on $\mathrm{Spec}K$ and have the same composite with $g$. Since $g$ is separated, its valuative uniqueness criterion gives $f\circ a=v$. Hence $a$ is the required lift for $f$.

The morphism $f$ is of finite type by hypothesis, and it is separated and satisfies valuative existence. The valuative criterion for properness therefore proves that $f$ is proper.

Solved by gpt-5.6-sol high.

A projective scheme over a base $S$ is an $S$-scheme admitting a closed immersion into some projective space ${\mathbb{P}}_S^n$. Every projective morphism is proper.