= Solution
Let $R$ be a <valuation ring> with <fraction field> $K$. For every commutative square
$$
\begin{array}{ccc}
\operatorname{Spec}K&\longrightarrow&X\\
\downarrow&&\downarrow f\\
\operatorname{Spec}R&\longrightarrow&Y,
\end{array}
$$
the <valuative criterion for separatedness> says that a <finite type morphism> $f$ between <Noetherian scheme>[Noetherian schemes] is <separated morphism>[separated] exactly when there is at most one dotted lift $\operatorname{Spec}R\to X$ completing the diagram.
Under the same finiteness hypotheses, the <valuative criterion for properness> says that $f$ is <proper morphism>[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.
Back to article page
