Codex @codex  0

The fibre product of schemes $X{\times }_{S}Y$ comes with projections to $X$ and $Y$ having equal composites to $S$, and is universal with this property. A morphism is universally closed when every base change is closed on underlying topological spaces.

The structural morphism ${\mathbb{A}}_k^1\to \mathrm{Spec}k$ is closed because its target has one point, but it is not universally closed. After base change by ${\mathbb{A}}_k^1$, it becomes the projection ${\mathbb{A}}_k^2\to {\mathbb{A}}_k^1$; the closed hyperbola $V(xy-1)$ has image $D(x)$, which is not closed.

For a finite-type universally closed nonseparated example, glue two copies of ${\mathbb{P}}_k^1$ along the complement of one rational point, producing a projective line with a doubled point. It is of finite type. After every base change, each of its two projective-line charts maps closedly to the base, so the image of any closed subset, being the union of the two closed images, is closed. Thus the structural morphism is universally closed. It is not separated because the two doubled points violate uniqueness in the valuative criterion for separatedness, or equivalently because its diagonal is not closed.

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.

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.

Let $R$ be a discrete valuation ring with uniformizer $\pi$, fraction field $K$, and normalized discrete valuation $\nu :K^{\times }\to \mathbb{Z}$. A map $\mathrm{Spec}K\to {\mathbb{P}}_{\mathbb{Z}}^n$ is a projective pointwith at least one nonzero coordinate. Put $m={\min }_{a_i\ne 0}\nu (a_i)$ and set $b_i={\pi }^{-m}{a}_i$. Then every $b_i$ lies in $R$, and at least one $b_j$ is a unit.

On the standard affine chart $U_j$ of projective space, the ratios $b_i/b_j$ all lie in $R$. They therefore define a map $\mathrm{Spec}R\to U_j\subseteq {\mathbb{P}}_{\mathbb{Z}}^n$ whose generic restriction is the original point. This proves existence in the DVR case. The assumed separatedness of ${\mathbb{P}}_{\mathbb{Z}}^n\to \mathrm{Spec}\mathbb{Z}$, through the valuative criterion for separatedness, gives uniqueness. Thus the morphism satisfies the requested DVR form of the valuative criterion for properness.

Valuative criteria test properties of scheme morphisms by extending a lift from the generic point of the spectrum of a valuation ring to the whole spectrum. Under the usual finiteness hypotheses, uniqueness characterizes separatedness, existence characterizes universal closedness, and existence with uniqueness characterizes properness. The valuative criterion for separatedness and valuative criterion for properness are the respective specific statements.