Examples
The projective space Pd over a field K is proper over a point (that is, Spec(K)). In the more classical language, this is the same as saying that projective space is a complete variety. Projective morphisms are proper, but not all proper morphisms are projective. Affine varieties of non-zero dimension are never proper. More generally, it can be shown that affine proper morphisms are necessarily finite. For example, it is not hard to see that the affine line A1 is not proper. In fact the map taking A1 to a point x is not universally closed. For example, the morphism
is not closed since the image of the hyperbola uv = 1, which is closed in A1 × A1, is the affine line minus the origin and thus not closed.
Read more about this topic: Proper Morphism
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