Valuative criterion

Remarks

The space $\operatorname{Spec} R$ has a generic point and a closed point, so it is an algebraic picture of a small curve. The space $\operatorname{Spec} K$ is that curve with its special point removed. Existence of a lift says that a curve in $X$ that is defined except at one point extends over that point, so limits exist; uniqueness says that limits are unique. For the proof, one realizes each specialization of points by a morphism from the spectrum of a valuation ring. This turns the closedness of images into the existence of lifts. For example, $\mathbb{A}^1_k \to \operatorname{Spec} k$ is not proper: with $R = k[t]_{(t)}$ and $u$ given by $x \mapsto 1/t$, there is no lift, since $1/t \notin R$. In $\mathbb{P}^1_k$ the same point is $[1 : 1/t] = [t : 1]$, which does extend over $t = 0$. If $Y$ is also Noetherian, then in both parts it suffices to use discrete valuation rings.

Going deeper

For a student

The definition of a proper morphism asks for universal closedness, which is hard to check directly. The valuative criterion replaces it by a lifting problem that one can test on explicit rings. It gives quick proofs that projective morphisms are proper and that properness is stable under composition and base change. It also underlies the construction of compact moduli spaces, where one shows that a family over a punctured curve has a unique limit. More general forms drop the Noetherian hypothesis and state the criterion for quasi-compact and quasi-separated morphisms.