Proper morphism

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

of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space

A morphism is finite if and only if it is proper and quasi-finite.

, 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]).

For any natural number n, projective space Pn 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.[1] 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.

[2] For example, it is not hard to see that the affine line A1 over a field k is not proper over k, because the morphism A1 → Spec(k) is not universally closed.

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 fraction field 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

More generally, a quasi-separated morphism f: X → 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 ∈ X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to

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.

For example, given the valuative criterion, it becomes easy to check that projective space Pn 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 [x0,...,xn] 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

Then, using a change of coordinates, this can be expressed as a power series on the unit disk.

Then, the valuative criterion for properness would be a filling in of the point

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.

There is another similar example of the valuative criterion of properness which captures some of the intuition for why this theorem should hold.

Then the valuative criterion for properness would read as a diagram

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.

be a morphism between locally noetherian formal schemes.

if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map

(EGA III, 3.4.1) The definition is independent of the choice of K. For example, if g: Y → Z is a proper morphism of locally noetherian schemes, Z0 is a closed subset of Z, and Y0 is a closed subset of Y such that g(Y0) ⊂ Z0, then the morphism

Grothendieck proved the coherence theorem in this setting.

be a proper morphism of locally noetherian formal schemes.