Proper map
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In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, a related concept is used.
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[edit] Topological spaces
[edit] Definition
A function f : X → Y between two topological spaces is proper if and only if the preimage of every compact set in Y is compact in X.
There are several competing descriptions. For instance, a continuous map f is proper if it is a closed map, and the pre-image of every point in Y is compact. More abstractly, f is proper if it is a closed map, and for any space Z the map (f, id Z) : X × Z → Y × Z is closed. These definitions are equivalent to the previous one if the space X is locally compact.
An equivalent, possibly more intuitive definition is as follows: We say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X, only finitely many points pi are in S. Then a map f : X → Y is proper if and only if, for every sequence of points {pi} that escapes to infinity in X, {f(pi)} escapes to infinity in Y.
[edit] Properties
- A topological space is compact if and only if the map from that space to a single point is proper.
- Every continuous map from a compact space to a Hausdorff space is both proper and closed.
- If f : X → Y is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces which are either first-countable or locally compact), then f is closed.
[edit] Algebraic varieties and schemes
[edit] Definition
A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber products f × Id: X × Z → Y × Z are closed maps of the underlying topological spaces. A morphism f : X → Y of algebraic varieties or is called proper if it is separated and universally closed. 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. A variety X over a field k is complete when the constant morphism from X to a point is proper.
A morphism f : X → Y of algebraic varieties over the field of complex numbers C induces a continuous function
between their sets of complex points with their complex topology (see GAGA). It can be shown that f is a proper morphism if and only if f(C) is a proper continuous function.
Properness is a local property that is stable under base change. The composition of two proper morphisms is proper.
[edit] 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. Closed immersions are proper. More generally, finite morphisms are proper.
Affine varieties of non-zero dimension are never proper. 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 f × Id: A1 × A1 → {x} × A1 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.
[edit] Valuative criterion of properness
There is a very intuitive criterion for properness which goes back to Chevalley. It is commonly called the valuative criterion of properness. Let f: X → Y be a morphism of finite type, with X noetherian. Then f is proper if and only if for all valuation rings R with fields of fractions K all K-valued points x ∈ X(K) that map to a point f(x) that is defined over R there is a unique lift of x to 
. (EGA II, 7.3.8 [2])
For example, the projective line is proper over a field (or even over Z) since one can always scale homogeneous co-ordinates by their least common denominator.
[edit] Stein factorization
A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism. (EGA III, 4.3.3 [3])
