(Proper)
A morphism is proper if it is separated, of finite type, and universally closed.
(Valuative criteria for valuation rings)
Let \(f \colon X \to Y\) be a quasiseparated morphism. For every valuation ring \(R\) with fraction field \(K\) and every commutative square \(\Spec K \to X\), \(\Spec R \to Y\) over \(f\), consider the lifts \(\Spec R \to X\) making both triangles commute.
\(f\) is separated if and only if every such square has at most one lift.
If \(f\) is of finite type, \(f\) is proper if and only if every such square has exactly one lift.
(Valuative criteria for discrete valuation rings)
Let \(f: X \to Y\) be of finite type with \(X\) Noetherian. For every discrete valuation ring \(R\) with fraction field \(K\) and every commutative square \[ \Spec K \to X, \qquad \Spec R \to Y , \] consider the lifts \(\Spec R \to X\) making both triangles commute.
\(f\) is separated if and only if every such square has at most one lift.
\(f\) is proper if and only if every such square has exactly one lift.
For a discrete valuation ring \(R\) with fraction field \(K\), \(\Spec R\) has one closed point and one generic point, like the germ of a smooth curve at a point, and \(\Spec K\) is the punctured germ. A lift \(\Spec R\to X\) extends a map of the punctured germ across the puncture. Uniqueness of extensions is the analogue of the Hausdorff property (limits of sequences are unique), and existence is the analogue of compactness (limits exist).
(A closed morphism that is not universally closed)
\(\AA^1_k \to \Spec k\) is separated, of finite type, and closed, but not proper: after base change along \(\AA^1_k\to\Spec k\), the projection \(\AA^1 \times \AA^1 \to \AA^1\) sends the closed hyperbola \(V(xy-1)\) onto \(\AA^1 \sm \ts{0}\), which is not closed.
Dependencies
D-T2J3QSeparated and quasi-separated morphismsuses
Backlinks
D-VARCOMPComplete varieties and projectivityusesPR-MORFINCHARFinite equals proper plus quasi-finiteusesD-MORPROJProjective and quasi-projective morphismsuses