Some authors call a proper variety over a field a complete variety. For example, every projective variety over a field is proper over . A scheme of finite type over the complex numbers (for example, a variety) is proper over if and only if the space of complex points with the classical (Euclidean) topology is compact.
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A morphism of schemes is called universally closed if for every scheme with a morphism , 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 is proper over . In particular, a variety over a field is said to be proper over if the morphism is proper.
Examples
For any natural number , projective space over a commutative ring is proper over . 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 .1Affine varieties of positive dimension over a field are never proper over . More generally, a proper affine morphism of schemes must be finite.2 For example, it is not hard to see that the affine line over a field is not proper over , because the morphism is not universally closed. Indeed, the pulled-back morphism
(given by ) is not closed, because the image of the closed subset in is , which is not closed in .
Properties and characterizations of proper morphisms
In the following, let be a morphism of schemes.
The composition of two proper morphisms is proper.
Any base change of a proper morphism is proper. That is, if is any morphism of schemes, then the resulting morphism is proper.
Properness is a local property on the base (in the Zariski topology). That is, if is covered by some open subschemes and the restriction of to all is proper, then so is .
More strongly, properness is local on the base in the fpqc topology. For example, if is a scheme over a field and is a field extension of , then is proper over if and only if the base change is proper over .3
For proper over a scheme , and separated over , the image of any morphism over is a closed subset of . This is analogous to the theorem in topology that the image of a continuous map from a compact space to a Hausdorff space is a closed subset.
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 be a morphism of finite type of Noetherian schemes. Then is proper if and only if for all discrete valuation rings with fraction field and for any -valued point that maps to a point that is defined over , there is a unique lift of to . (EGA II, 7.3.8). More generally, a quasi-separated morphism of finite type (note: finite type includes quasi-compact) of 'any' schemes , is proper if and only if for all valuation rings with fraction field and for any -valued point that maps to a point that is defined over , there is a unique lift of to . (Stacks project Tags 01KF and 01KY). Noting that is the generic point of and discrete valuation rings are precisely the regularlocal one-dimensional rings, one may rephrase the criterion: given a regular curve on (corresponding to the morphism ) and given a lift of the generic point of this curve to , is proper if and only if there is exactly one way to complete the curve.
Similarly, is separated if and only if in every such diagram, there is at most one lift .
For example, given the valuative criterion, it becomes easy to check that projective space is proper over a field (or even over ). One simply observes that for a discrete valuation ring with fraction field , every -point of projective space comes from an -point, by scaling the coordinates so that all lie in and at least one is a unit in .
Geometric interpretation with disks
One of the motivating examples for the valuative criterion of properness is the interpretation of as an infinitesimal disk, or complex-analytically, as the disk . This comes from the fact that every power series
Proper morphism of formal schemes
Let be a morphism between locally noetherian formal schemes. We say is proper or is proper over if (i) is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map is proper, where and is the ideal of definition of .(EGA III, 3.4.1) The definition is independent of the choice of .
For example, if is a proper morphism of locally noetherian schemes, is a closed subset of , and is a closed subset of such that , then the morphism on formal completions is a proper morphism of formal schemes.
Grothendieck proved the coherence theorem in this setting. Namely, let be a proper morphism of locally noetherian formal schemes. If is a coherent sheaf on , then the higher direct images are coherent.11
The Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factored as , where is proper, surjective, and has geometrically connected fibers, and is finite.7
Chow's lemma says that proper morphisms are closely related to projective morphisms. One version is: if is proper over a quasi-compact scheme and has only finitely many irreducible components (which is automatic for noetherian), then there is a projective surjective morphism such that is projective over . Moreover, one can arrange that is an isomorphism over a dense open subset of , and that is dense in . One can also arrange that is integral if is integral.8
Nagata's compactification theorem, as generalized by Deligne, says that a separated morphism of finite type between quasi-compact and quasi-separated schemes factors as an open immersion followed by a proper morphism.9
Proper morphisms between locally noetherian schemes preserve coherent sheaves, in the sense that the higher direct images (in particular the direct image ) of a coherent sheaf are coherent (EGA III, 3.2.1). (Analogously, for a proper map between complex analytic spaces, Grauert and Remmert showed that the higher direct images preserve coherent analytic sheaves.) As a very special case: the ring of regular functions on a proper scheme over a field has finite dimension as a -vector space. By contrast, the ring of regular functions on the affine line over is the polynomial ring , which does not have finite dimension as a -vector space.
There is also a slightly stronger statement of this:(EGA III, 3.2.4) let be a morphism of finite type, locally noetherian and a -module. If the support of is proper over , then for each the higher direct image is coherent.
For a scheme of finite type over the complex numbers, the set of complex points is a complex analytic space, using the classical (Euclidean) topology. For and separated and of finite type over , a morphism over is proper if and only if the continuous map is proper in the sense that the inverse image of every compact set is compact.10
If and are such that is proper and is separated, then is proper. This can for example be easily proven using the following criterion.
converges in some disk of radius around the origin. Then, using a change of coordinates, this can be expressed as a power series on the unit disk. Then, if we invert , this is the ring which are the power series which may have a pole at the origin. This is represented topologically as the open disk with the origin removed. For a morphism of schemes over , this is given by the commutative diagram
Then, the valuative criterion for properness would be a filling in of the point in the image of .
Example
It's instructive to look at a counter-example to see why the valuative criterion of properness should hold on spaces analogous to closed compact manifolds. If we take and , then a morphism factors through an affine chart of , reducing the diagram to
where is the chart centered around on . This gives the commutative diagram of commutative algebras
Then, a lifting of the diagram of schemes, , would imply there is a morphism sending from the commutative diagram of algebras. This, of course, cannot happen. Therefore is not proper over .
Geometric interpretation with curves
There is another similar example of the valuative criterion of properness which captures some of the intuition for why this theorem should hold. Consider a curve and the complement of a point . Then the valuative criterion for properness would read as a diagram
with a lifting of . Geometrically this means every curve in the scheme can be completed to a compact curve. This bit of intuition aligns with what the scheme-theoretic interpretation of a morphism of topological spaces with compact fibers, that a sequence in one of the fibers must converge. Because this geometric situation is a problem locally, the diagram is replaced by looking at the local ring , which is a DVR, and its fraction field . Then, the lifting problem then gives the commutative diagram
where the scheme represents a local disk around with the closed point removed.
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Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.
Algebraic structure with addition, multiplication, and division
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do.
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In mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set.
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In mathematics, particularly topology, the tube lemma, also called Wallace's theorem, is a useful tool in order to prove that the product of finitely many compact spaces is compact.
In algebraic geometry, a branch of mathematics, a morphism f: X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions: Every point x of X is isolated in its fiber f−1(f(x)).
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In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
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