Proper morphism explained

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 k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(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.

Definition

A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product

X x _{Y} Z \to Z 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 X is proper over Y. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper.

Examples

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. Indeed, the pulled-back morphism

A ^{1} x _{k} A ^{1} \to A ^{1} (given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A¹ × A¹ = A² is A¹ − 0, which is not closed in A¹.

Properties and characterizations of proper morphisms

In the following, let f: X → Y be a morphism of schemes.

• The composition of two proper morphisms is proper.
• Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×_Y Z → Z is proper.
• Properness is a local property on the base (in the Zariski topology). That is, 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.
• More strongly, properness is local on the base in the fpqc topology. For example, if X is a scheme over a field k and E is a field extension of k, then X is proper over k if and only if the base change X_E is proper over E.^[3]
• Closed immersions are proper.
• More generally, finite morphisms are proper. This is a consequence of the going up theorem.
• By Deligne, a morphism of schemes is finite if and only if it is proper and quasi-finite.^[4] This had been shown by Grothendieck if the morphism f: X → Y is locally of finite presentation, which follows from the other assumptions if Y is noetherian.^[5]
• For X proper over a scheme S, and Y separated over S, the image of any morphism X → Y over S is a closed subset of Y.^[6] 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.
• The Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factored as X → Z → Y, where X → Z is proper, surjective, and has geometrically connected fibers, and Z → Y is finite.^[7]
• Chow's lemma says that proper morphisms are closely related to projective morphisms. One version is: if X is proper over a quasi-compact scheme Y and X has only finitely many irreducible components (which is automatic for Y noetherian), then there is a projective surjective morphism g: W → X such that W is projective over Y. Moreover, one can arrange that g is an isomorphism over a dense open subset U of X, and that g⁻¹(U) is dense in W. One can also arrange that W is integral if X 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 Rⁱf_∗(F) (in particular the direct image f_∗(F)) of a coherent sheaf F 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 X over a field k has finite dimension as a k-vector space. By contrast, the ring of regular functions on the affine line over k is the polynomial ring k[''x''], which does not have finite dimension as a k-vector space.
• There is also a slightly stronger statement of this: let

f\colon X \to S be a morphism of finite type, S locally noetherian and F a l{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.

• For a scheme X of finite type over the complex numbers, the set X(C) of complex points is a complex analytic space, using the classical (Euclidean) topology. For X and Y separated and of finite type over C, a morphism f: X → Y over C is proper if and only if the continuous map f: X(C) → Y(C) is proper in the sense that the inverse image of every compact set is compact.
• 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

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

\overline{x} \in X(R) . (EGA II, 7.3.8). 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 \overline{x} \in X(R) . (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 → 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

\overline{x} \in X(R) .

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 [''x''₀,...,''x''_''n''] 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.

Geometric interpretation with disks

One of the motivating examples for the valuative criterion of properness is the interpretation of

Spec ( C [[t]]) as an infinitesimal disk, or complex-analytically, as the disk \Delta = \{x \in C : |x| < 1 \} . This comes from the fact that every power series

f(t) = \sum ^{inf ty} _{n =0} a ^{n} _{nt}

converges in some disk of radius r 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 t , this is the ring C [[t]][t ^{-1} ] = C ((t)) which are the power series which may have a pole at the origin. This is represented topologically as the open disk \Delta ^{*} = \{x \in C : 0 <|x| < 1 \} with the origin removed. For a morphism of schemes over Spec ( C ) , this is given by the commutative diagram

\begin{matrix} \Delta ^{*} & \to & X \\ \downarrow & & \downarrow \\ \Delta & \to & Y \end{matrix}

Then, the valuative criterion for properness would be a filling in of the point 0 \in \Delta in the image of \Delta ^{*} .

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

X = P ^{1} - \{x \} and Y = Spec ( C ) , then a morphism Spec ( C ((t))) \to X factors through an affine chart of X , reducing the diagram to

\begin{matrix} Spec ( C ((t))) & \to & Spec ( C [t,t ^{-1} ]) \\ \downarrow & & \downarrow \\ Spec ( C [[t]]) & \to & Spec ( C ) \end{matrix}

where Spec ( C [t,t ^{-1} ]) = A ^{1} - \{ 0 \} is the chart centered around \{x \} on X . This gives the commutative diagram of commutative algebras

\begin{matrix} C ((t)) & \leftarrow & C [t,t ^{-1} ] \\ \uparrow & & \uparrow \\ C [[t]] & \leftarrow & C \end{matrix}

Then, a lifting of the diagram of schemes, Spec ( C [[t]]) \to Spec ( C [t,t ^{-1} ]) , would imply there is a morphism C [t,t ^{-1} ] \to C [[t]] sending t \mapsto t from the commutative diagram of algebras. This, of course, cannot happen. Therefore X is not proper over Y .

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

C and the complement of a point C - \{p\} . Then the valuative criterion for properness would read as a diagram

\begin{matrix} C - \{p\} & → & X \\ \downarrow & & \downarrow \\ C & → & Y \end{matrix}

with a lifting of C \to X . Geometrically this means every curve in the scheme X 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 l{O} _{C,ak{p} }, which is a DVR, and its fraction field Frac ( l{O} _{C,ak{p} }). Then, the lifting problem then gives the commutative diagram

\begin{matrix} Spec ( Frac ( l{O} _{C,ak{p} })) & \rightarrow & X \\\downarrow & & \downarrow \\\text(\mathcal_) & \rightarrow & Y\end

where the scheme Spec ( Frac ( l{O} _{C,ak{p} })) represents a local disk around ak{p} with the closed point ak{p} removed.

Proper morphism of formal schemes

Let

f\colon ak{X} \to ak{S} be a morphism between locally noetherian formal schemes. We say f is proper or ak{X} is proper over ak{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\colon} X _{0} \to S _{0} is proper, where X _{0} = ( ak{X}, l{O} _{ak{X}/I),S} _{0} = ( ak{S}, l{O} _{ak{S}/K),I} = f ^{*(K)} l{O} _{ak{X}} and K is the ideal of definition of ak{S} . 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

\widehat{g}\colon Y _{/Y _{0}} \to Z _{/Z _{0}} on formal completions is a proper morphism of formal schemes.

Grothendieck proved the coherence theorem in this setting. Namely, let

f\colon ak{X} \to ak{S} be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on ak{X} , then the higher direct images R ^{i} f _{*} F are coherent.^[10]

See also

• Proper base change theorem
• Stein factorization

References

• SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971
  • , section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
• , section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)

Notes and References

1. Hartshorne (1977), Appendix B, Example 3.4.1.
2. Liu (2002), Lemma 3.3.17.
3. .
4. Grothendieck, EGA IV, Part 4, Corollaire 18.12.4; .
5. Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
6. .
7. .
8. Grothendieck, EGA II, Corollaire 5.6.2.
9. Conrad (2007), Theorem 4.1.
10. Grothendieck, EGA III, Part 1, Théorème 3.4.2.