# Proper morphism

In algebraic geometry, a proper morphism between schemes is a scheme-theoretic analogue of a proper map between complex-analytic varieties.

A basic example is a complete variety (e.g., projective variety) in the following sense: a k-variety X is complete in the classical definition if it is universally closed. A proper morphism is a generalization of this to schemes.

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

## Definition

A morphism f : XY of algebraic varieties or more generally of schemes, is called universally closed if for all morphisms ZY, the projections for the fiber product

${\displaystyle X\times _{Y}Z\to Z}$

are closed maps of the underlying topological spaces. A morphism f : XY 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 structural morphism from X to the spectrum of k is proper.

## Examples

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. For example, it can be shown that the scheme obtained by contracting two disjoint projective lines in some P3 to one is a proper, but non-projective variety.[1] Affine varieties of non-zero dimension are never complete. More generally, it can be shown that affine proper morphisms are necessarily finite.[2] For example, it is not hard to see that the affine line A1 is not complete. In fact the map taking A1 to a point x is not universally closed. For example, the morphism

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

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 : XY be a morphism of schemes.

between their sets of complex points with their complex topology. (This is an instance of GAGA.) Then f is a proper morphism defined above if and only if ${\displaystyle f(\mathbb {C} )}$ is a proper map in the sense of Bourbaki and is separated.[4]
• If f: XY and g:YZ 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: XY 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 xX(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to ${\displaystyle {\overline {x}}\in X(R)}$. (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 ${\displaystyle {\overline {x}}\in X(R)}$.

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.

## Proper morphism of formal schemes

Let ${\displaystyle f:{\mathfrak {X}}\to {\mathfrak {S}}}$ be a morphism between locally noetherian formal schemes. We say f is proper or ${\displaystyle {\mathfrak {X}}}$ is proper over ${\displaystyle {\mathfrak {S}}}$ if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map ${\displaystyle f_{0}:X_{0}\to Y_{0}}$ is proper, where ${\displaystyle X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}}$ and K is the ideal of definition of ${\displaystyle {\mathfrak {S}}}$.Template:Harv The definition is independent of the choice of K. If one lets ${\displaystyle X_{n}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I^{n+1}),S_{n}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/K^{n+1})}$, then${\displaystyle f_{n}:X_{n}\to S_{n}}$ is proper.

For example, if ${\displaystyle g:Y\to Z}$ is a proper morphism, then its extension ${\displaystyle {\widehat {g}}:{\widehat {Y}}\to {\widehat {Z}}}$ between formal completions is proper in the above sense.

As before, we have the coherence theorem: let ${\displaystyle f:{\mathfrak {X}}\to {\mathfrak {S}}}$ be a proper morphism between locally noetherian formal schemes. If F is a coherent ${\displaystyle {\mathcal {O}}_{\mathfrak {X}}}$-module, then the higher direct images ${\displaystyle R^{i}f_{*}F}$ are coherent.

## References

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