Valuative criterion of properness

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.

Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.

A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.

A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product

is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ([EGA] II, 5.4.1 [1]). One also says that X is proper over Y. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper.

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