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 <math>k</math> a complete variety. For example, every projective variety over a field <math>k</math> is proper over <math>k</math>. A scheme <math>X</math> of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space <math>X</math>(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 <math>f:X\to Y</math> of schemes is called universally closed if for every scheme <math>Z</math> with a morphism <math>Z\to Y</math>, the projection from the fiber product

<math>X \times_Y Z \to Z</math>

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]). One also says that <math>X</math> is proper over <math>Y</math>. In particular, a variety <math>X</math> over a field <math>k</math> is said to be proper over <math>k</math> if the morphism <math>X\to\operatorname{Spec}(k)</math> 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.<ref>Hartshorne (1977), Appendix B, Example 3.4.1.</ref> 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.<ref>Liu (2002), Lemma 3.3.17.</ref> 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

<math>\mathbb{A}^1 \times_k \mathbb{A}^1 \to \mathbb{A}^1</math>

(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.<ref>Template:Citation.</ref>
• 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.<ref>Grothendieck, EGA IV, Part 4, Corollaire 18.12.4; Template:Citation.</ref> 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.<ref>Grothendieck, EGA IV, Part 3, Théorème 8.11.1.</ref>
• 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.<ref>Template:Citation.</ref> 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.<ref>Template:Citation.</ref>
• 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.<ref>Grothendieck, EGA II, Corollaire 5.6.2.</ref>
• 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.<ref>Conrad (2007), Theorem 4.1.</ref>
• 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:Template:Harv let <math>f\colon X \to S</math> be a morphism of finite type, S locally noetherian and <math>F</math> a <math>\mathcal{O}_X</math>-module. If the support of F is proper over S, then for each <math>i \ge 0</math> the higher direct image <math>R^i f_* F</math> 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.<ref>Template:Harvnb</ref>
• 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.

File:Valuative criterion of properness.png

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: 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 <math>\overline{x} \in X(R)</math>. (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 <math>\overline{x} \in X(R)</math>. (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 <math>\overline{x} \in X(R)</math>.

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

<math>f(t) = \sum_{n=0}^\infty a_nt^n</math>

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

<math>\begin{matrix}

\Delta^* & \to & X \\ \downarrow & & \downarrow \\ \Delta & \to & Y

\end{matrix}</math>

Then, the valuative criterion for properness would be a filling in of the point <math>0 \in \Delta</math> in the image of <math>\Delta^*</math>.

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

<math>\begin{matrix}

\text{Spec}(\mathbb{C}((t))) & \to & \text{Spec}(\mathbb{C}[t,t^{-1}]) \\ \downarrow & & \downarrow \\ \text{Spec}(\mathbb{C}t) & \to & \text{Spec}(\mathbb{C})

\end{matrix}</math>

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

<math>\begin{matrix}

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

\end{matrix}</math>

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

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

<math>\begin{matrix}

C-\{p\} & \rightarrow & X \\ \downarrow & & \downarrow \\ C & \rightarrow & Y

\end{matrix}</math>

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

<math>\begin{matrix}

\text{Spec}(\text{Frac}(\mathcal{O}_{C,\mathfrak{p}}) ) & \rightarrow & X \\ \downarrow & & \downarrow \\ \text{Spec}(\mathcal{O}_{C,\mathfrak{p}} ) & \rightarrow & Y

\end{matrix}</math>

where the scheme <math>\text{Spec}(\text{Frac}(\mathcal{O}_{C,\mathfrak{p}}))</math> represents a local disk around <math>\mathfrak{p}</math> with the closed point <math>\mathfrak{p}</math> removed.

Proper morphism of formal schemes

Let <math>f\colon \mathfrak{X} \to \mathfrak{S}</math> be a morphism between locally noetherian formal schemes. We say f is proper or <math>\mathfrak{X}</math> is proper over <math>\mathfrak{S}</math> if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map <math>f_0\colon X_0 \to S_0</math> is proper, where <math>X_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I), S_0 = (\mathfrak{S}, \mathcal{O}_\mathfrak{S}/K), I = f^*(K) \mathcal{O}_\mathfrak{X}</math> and K is the ideal of definition of <math>\mathfrak{S}</math>.Template:Harv 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 <math>\widehat{g}\colon Y_{/Y_0} \to Z_{/Z_0}</math> on formal completions is a proper morphism of formal schemes.

Grothendieck proved the coherence theorem in this setting. Namely, let <math>f\colon \mathfrak{X} \to \mathfrak{S}</math> be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on <math>\mathfrak{X}</math>, then the higher direct images <math>R^i f_* F</math> are coherent.<ref>Grothendieck, EGA III, Part 1, Théorème 3.4.2.</ref>

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
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• Template:EGA, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
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• Template:EGA, section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)
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External links

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