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.

A morphism ${\displaystyle f:X\to Y}$ of schemes is called universally closed if for every scheme ${\displaystyle Z}$ with a morphism ${\displaystyle Z\to Y}$, 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 ${\displaystyle X}$ is proper over ${\displaystyle Y}$. In particular, a variety ${\displaystyle X}$ over a field ${\displaystyle k}$ is said to be proper over ${\displaystyle k}$ if the morphism ${\displaystyle X\to \operatorname {Spec} (k)}$ is proper.

For any natural number n, projective space Pⁿ over a commutative ring R is proper over R. Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a smooth proper complex variety of dimension 3 which is not projective over C. Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite. For example, it is not hard to see that the affine line A¹ over a field k is not proper over k, because the morphism A¹ → Spec(k) is not universally closed. Indeed, the pulled-back morphism

(given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A¹ × A¹ = A² is A¹ − 0, which is not closed in A¹.

In the following, let f: X → Y be a morphism of schemes.