In algebraic geometry
, a proper morphism between schemes
is an analogue of a proper map
between topological spaces
A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber products
are closed maps
of the underlying topological spaces. A morphism f
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
). One also says that X
is proper over Y
. A variety X
over a field k
when the constant morphism from X
to a point is proper.
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.
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
, is the affine line minus the origin and thus not closed.
Properties and characterizations of proper morphisms
In the following, let f
be a morphism of varieties or schemes.
- If f is defined over the field of complex numbers C, it 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 on the base, i.e. if Y is covered by some open subschemes Yi and the restriction of f to all f-1(Yi) is proper, then so is f.
- By definition, proper morphisms are stable under base change.
- The composition of two proper morphisms is proper.
- By definition, closed immersions are proper.
- More generally, finite morphisms are proper. This is a consequence of the going up theorem.
- Conversely, every quasi-finite 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' := f ○ g is projective. This statement is called Chow's lemma.
- Proper 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 . Its sections is the ring of Laurent polynomials, which is infinitely generated over C.
- If f: X→Y and g:Y→Z are such that g○f 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
There is a very intuitive criterion for properness which goes back to Chevalley
. It is commonly called the valuative criterion of properness
. Let f
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
) that maps to a point f
) that is defined over R
, there is a unique lift of x
. (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
is proper iff
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 .
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.
- , section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
- , section 15.7. (generalisations of valuative criteria to not necessarily noetherian schemes)