= Valuative criterion for universal closedness
Among <quasi-compact morphisms> of <schemes>, <universal closedness> is equivalent to the existence part of the <valuative criterion>: for every <valuation ring> $D$ with <fraction field> $K$, every compatible pair $\operatorname{Spec}K\to X$, $\operatorname{Spec}D\to S$ extends to $\operatorname{Spec}D\to X$. All <valuation rings> must be tested without further hypotheses. A <universally closed morphism> is automatically a <quasi-compact morphism>.
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