Proper morphism

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 ${\displaystyle k}$ a complete variety. For example, every projective variety over a field ${\displaystyle k}$ is proper over ${\displaystyle k}$. A scheme ${\displaystyle X}$ of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space ${\displaystyle X}$(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.