Proper morphism

In algebraic geometry, a proper morphism between schemes is an analogue of a proper map between topological spaces.

Definition

A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber products

$f \times \textrm{id}: X \times Z \to Y \times Z$

are closed maps of the underlying topological spaces. A morphism f : X → Y of algebraic varieties 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.

Examples

The projective space P^d 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

$f \times \textrm{id}: \mathbb{A}^1 \times \mathbb{A}^1 \to \{x\} \times \mathbb{A}^1$

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.

Properties and characterizations of proper morphisms

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

• If X, Y are varieties over the field of complex numbers C, it induces a continuous function

$f(\mathbf{C}): X(\mathbf{C}) \to Y(\mathbf{C})$

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 on the base, i.e. if Y is covered by some open subschemes Y_i and the restriction of f to all f^-1(Y_i) is proper, then so is f.
• By definition, proper morphisms are stable under base change.
• The composition of two proper morphisms is proper.
• Closed immersions are proper.
• More generally, finite morphisms are proper. This is a consequence of the going up theorem.
• Conversely, every quasi-finite and finite presentation proper morphism is finite. This follows from the so-called 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) This is a deep theorem.
• Proper morphisms are closely related to projective morphisms: If f is proper over a noetherian base Y, then there is a morphism: g: X' →X which is an isomorphism when restricted to a suitable open dense subset: g^-1(U) ≅ U, such that f'  := fg is projective. This statement is called Chow's lemma.

• Proper finite presentation morphisms of schemes or complex analytic spaces preserve coherent sheaves, in the sense that the higher direct images Rⁱf_∗(F) (in particular the direct image f_∗(F)) of a coherent sheaf F are coherent (EGA III, 3.2.1). This boils down to the fact that the cohomology groups of projective space over some field k with respect to coherent sheaves are finitely generated over k, a statement which fails for non-projective varieties: consider C^∗, the punctured disc and its sheaf of holomorphic functions $\mathcal O$. Its sections $\mathcal O(\mathbb C^*)$ is the ring of Laurent polynomials, which is infinitely generated over C.
• If f: X→Y and g:Y→Z are such that gf is proper and g is separated, then f is proper. This can for example be easily proven using the following criterion:

Valuative criterion of properness

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 of noetherian schemes. Then f is proper if and only if for all discrete valuation rings R with fields of fractions K and for any K-valued point x ∈ X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to $\overline{x} \in X(R)$. (EGA II, 7.3.8). Noting that Spec K is the generic point of Spec R and discrete valuation rings are precisely the regular local one-dimensional rings, one may rephrase the criterion: given a regular curve on Y (corresponding to the morphism s : Spec R → Y) and given a lift of the generic point of this curve to X, f is proper if and only if there is exactly one way to complete the curve.

Similarly, f is separated if and only if in all such diagrams, there is at most one lift $\overline{x} \in X(R)$.

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.

See also

• Glossary of scheme theory

References

• Grothendieck, Alexandre; Dieudonné, Jean (1961), "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes", Publications Mathématiques de l'IHÉS 8: 5–222, doi:10.1007/BF02699291, ISSN 1618-1913, http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_ , section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
• Grothendieck, Alexandre; Dieudonné, Jean (1966), "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie", Publications Mathématiques de l'IHÉS 28: 5–255, ISSN 1618-1913, http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_ , section 15.7. (generalisations of valuative criteria to not necessarily noetherian schemes)
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR0463157 

External links

• V.I. Danilov (2001), "Proper morphism", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/P/p075450.htm