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 : X → Y of algebraic varieties or more generally of schemes, is called universally closed if for all morphisms Z → Y, the projections for the fiber product

$X \times_Y Z \to Z$

are closed maps of the underlying topological spaces. A morphism f : X → Y 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 P^d 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. Affine varieties of non-zero dimension are never complete. 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 complete. In fact the map taking A¹ to a point x is not universally closed. For example, the morphism

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

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 : X → Y be a morphism of schemes.

• Properness is a local property on the base, i.e. if Y is covered by some open subschemes Y_i and the restriction of f to all f⁻¹(Y_i) is proper, then so is f.
• Proper morphisms are stable under base change and composition.
• Closed immersions are proper.
• More generally, finite morphisms are proper. This is a consequence of the going up theorem.
• Conversely, every quasi-finite, locally of finite presentation and proper morphism is finite. (EGA III, 4.4.2 in the noetherian case and EGA IV, 8.11.1 for the general case)
• Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factorized into $X\to Z\to Y$, where the first morphism has geometrically connected fibers and the second one is finite.
• 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'  := fg is projective. This statement is called Chow's lemma.
• Nagata's compactification theorem says that a separated morphism of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism.
• Proper morphisms between locally noetherian 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 $\mathcal O$. Its sections $\mathcal O(\mathbb C^*)$ is the ring of Laurent polynomials, which is infinitely generated over C.
• There is also a slightly stronger statement of this:(EGA III, 3.2.4) let $f: X \to S$ be a morphism of finite type, S locally noetherian and $F$ a $\mathcal{O}_X$-module. If the support of F is proper over S, then for each $i \ge 0$ the higher direct image $R^i f_* F$ is coherent.:
• (SGA 1, XII) If X, Y are schemes of locally of finite type over the field of complex numbers $\mathbb{C}$, f induces a morphism of complex analytic spaces

  $f(\mathbb{C}): X(\mathbb{C}) \to Y(\mathbb{C})$

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 $f(\mathbb{C})$ is a proper map in the sense of Bourbaki and is separated.

• If f: X→Y and g:Y→Z 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

File:Valuative criterion of properness.png

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: 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 fields of fractions 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 $\overline{x} \in X(R)$. (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 $\overline{x} \in X(R)$.

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 $f: \mathfrak{X} \to \mathfrak{S}$ be a morphism between locally noetherian formal schemes. We say f is proper or $\mathfrak{X}$ is proper over $\mathfrak{S}$ if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map $f_0: X_0 \to Y_0$ is proper, where $X_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I), S_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/K), I = f^*(K) \mathcal{O}_\mathfrak{X}$ and K is the ideal of definition of $\mathfrak{S}$.(EGA III, 3.4.1) The definition is independent of the choice of K. If one lets $X_n = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I^{n+1}), S_n = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/K^{n+1})$, then$f_n: X_n \to S_n$ is proper.

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

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

See also

• Proper base change theorem
• Stein factorization

References

• , 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)