Past exam of the introduction to schemes course of the University of Oxford / 2023 / 2 / a

For every valuation ring $D$ with fraction field $K$, consider compatible morphisms of schemesAn extension is a morphism of schemes $\ell :\mathrm{Spec}D\to X$ satisfying $\ell {|}_{\mathrm{Spec}K}=u$ and $f\ell =v$. The valuative criterion for universal closedness statesEquivalently, among quasi-compact morphisms, the existence condition characterizes universal closedness. A universally closed morphism is automatically a quasi-compact morphism; the converse direction of the criterion needs this compactness condition. All valuation rings are allowed, with arbitrary fraction fields; there is no Noetherian ring hypothesis and one cannot replace them by discrete valuation rings without additional assumptions.^[ref][ref]

Use the compatible pair $u,v$ and the notion of extension from (i). The valuative criterion for separatedness isA quasi-separated morphism means one whose diagonal morphism is a quasi-compact morphism. A separated morphism automatically has this property because its diagonal morphism is a closed immersion. The quasi-separated morphism hypothesis is needed in the converse valuative test for arbitrary schemes. Existence is not demanded.^[ref]

With the same compatible pair $u,v$, the valuative criterion for properness isEquivalently, one can replace finite type by being a quasi-compact morphism and locally of finite type. Necessity follows from the definition of a proper morphism and the existence and uniqueness criteria in (i) and (ii). Conversely, finite type supplies the compactness needed in (i), while being a quasi-separated morphism supplies the compactness of the diagonal morphism needed in (ii). Thus the extension condition gives both universal closedness and separatedness, and these together with finite type give a proper morphism. These hypotheses are part of the criteria, even though the question is phrased for unrestricted morphisms of schemes.^[ref]