In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
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 C if and only if the space (C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.
A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.
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 https://web.archive.org/web/20051108184937/http://modular.fas.harvard.edu/scans/papers/grothendieck/PMIHES_1961__8__5_0.pdf). 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.
For any natural number n, projective space P<sup>n</sup> over a commutative ring R is proper over R. 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 C. Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite. For example, it is not hard to see that the affine line A<sup>1</sup> over a field k is not proper over k, because the morphism A<sup>1</sup> â Spec(k) is not universally closed. Indeed, the pulled-back morphism
(given by (x,y) ⦠y) is not closed, because the image of the closed subset xy = 1 in A<sup>1</sup> àA<sup>1</sup> = A<sup>2</sup> is A<sup>1</sup> â 0, which is not closed in A<sup>1</sup>.
In the following, let f: X â Y be a morphism of schemes.
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 â 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 fraction field K and for any K-valued point x â X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to . (EGA II, 7.3.8). More generally, a quasi-separated morphism f: X â Y of finite type (note: finite type includes quasi-compact) of 'any' schemes X, Y is proper if and only if for all valuation rings R with fraction field K and for any K-valued point x â X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to . (Stacks project Tags 01KF and 01KY). 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 â 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 every such diagram, there is at most one lift .
For example, given the valuative criterion, it becomes easy to check that projective space P<sup>n</sup> is proper over a field (or even over Z). One simply observes that for a discrete valuation ring R with fraction field K, every K-point [x<sub>0</sub>,...,x<sub>n</sub>] of projective space comes from an R-point, by scaling the coordinates so that all lie in R and at least one is a unit in R.
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<blockquote></blockquote>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<blockquote></blockquote>Then, the valuative criterion for properness would be a filling in of the point in the image of .
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<blockquote></blockquote>where is the chart centered around on . This gives the commutative diagram of commutative algebras<blockquote></blockquote>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 .
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<blockquote></blockquote>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<blockquote></blockquote>where the scheme represents a local disk around with the closed point removed.
Let be a morphism between locally noetherian formal schemes. We say f is proper or is proper over if (i) f 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 K is the ideal of definition of . The definition is independent of the choice of K.
For example, if g: Y â Z is a proper morphism of locally noetherian schemes, Z<sub>0</sub> is a closed subset of Z, and Y<sub>0</sub> is a closed subset of Y such that g(Y<sub>0</sub>) â Z<sub>0</sub>, 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 F is a coherent sheaf on , then the higher direct images are coherent.