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

: $X \times_Y Z \to Z$

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 $X\to\operatorname{Spec}(k)$ 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.Hartshorne (1977), Appendix B, Example 3.4.1. 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.Liu (2002), Lemma 3.3.17. 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

: $\mathbb{A}^1 \times_k \mathbb{A}^1 \to \mathbb{A}^1$

(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.{{Citation | title=Stacks Project, Tag 02YJ | url=http://stacks.math.columbia.edu/tag/02YJ}}.
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.Grothendieck, EGA IV, Part 4, Corollaire 18.12.4; {{Citation | title=Stacks Project, Tag 02LQ | url=http://stacks.math.columbia.edu/tag/02LQ}}. 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.Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
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.{{Citation | title=Stacks Project, Tag 01W0 | url=http://stacks.math.columbia.edu/tag/01W0}}. 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.{{Citation | title=Stacks Project, Tag 03GX | url=http://stacks.math.columbia.edu/tag/03GX}}.
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.Grothendieck, EGA II, Corollaire 5.6.2.
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.Conrad (2007), Theorem 4.1.
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:{{harv|EGA III|loc=3.2.4}} let $f\colon X \to S$ 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.
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.{{harvnb|SGA 1|loc=XII Proposition 3.2.}}
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.

Image:Valuative criterion of properness.png 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 $\overline{x} \in X(R)$ . (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 [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 $\text{Spec}(\mathbb{C} t)$ as an infinitesimal disk, or complex-analytically, as the disk $\Delta = \{x \in \mathbb{C} : |x| < 1 \}$ . This comes from the fact that every power series

$f(t) = \sum_{n=0}^\infty a_nt^n$

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

[t^{-1}] = \mathbb{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 \mathbb{C} : 0<|x| < 1 \}

with the origin removed. For a morphism of schemes over

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

$) & \to & \text{Spec}(\mathbb{C}) \end{matrix}$

where

\text{Spec}(\mathbb{C}[t,t^{-1}]) = \mathbb{A}^1 - \{0\}

is the chart centered around

\{x \}

X

. This gives the commutative diagram of commutative algebras

$& \leftarrow & \mathbb{C} \end{matrix}$

Then, a lifting of the diagram of schemes,

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

, would imply there is a morphism

\mathbb{C}[t,t^{-1}] \to \mathbb{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

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

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

\mathcal{O}_{C,\mathfrak{p}}

, which is a DVR, and its fraction field

\text{Frac}(\mathcal{O}_{C,\mathfrak{p}})

. Then, the lifting problem then gives the commutative diagram

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

where the scheme

\text{Spec}(\text{Frac}(\mathcal{O}_{C,\mathfrak{p}}))

represents a local disk around

\mathfrak{p}

with the closed point

\mathfrak{p}

removed.

Proper morphism of formal schemes

Let $f\colon \mathfrak{X} \to \mathfrak{S}$ 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 $f_0\colon X_0 \to S_0$ is proper, where $X_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I), S_0 = (\mathfrak{S}, \mathcal{O}_\mathfrak{S}/K), I = f^*(K) \mathcal{O}_\mathfrak{X}$ and K is the ideal of definition of $\mathfrak{S}$ .{{harv|EGA III|loc=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 $\widehat{g}\colon Y_{/Y_0} \to Z_{/Z_0}$ on formal completions is a proper morphism of formal schemes.

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

References

SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971
{{Citation | author1-last=Conrad | author1-first=Brian | author1-link=Brian Conrad | title=Deligne's notes on Nagata compactifications | journal=Journal of the Ramanujan Mathematical Society | volume=22 | year=2007 | pages=205–257 | mr=2356346 | url=http://math.stanford.edu/~conrad/papers/nagatafinal.pdf}}
{{EGA|book=2| pages = 5–222}}, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
{{EGA|book=3-1| pages = 5–167}}
{{EGA|book=4-3| pages = 5–255}}, section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)
{{EGA|book=4-4| pages = 5–361}}
{{Citation | last1=Hartshorne | first1=Robin | author1-link= Robin Hartshorne | title=Algebraic Geometry | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90244-9 |mr=0463157 | year=1977}}
{{Citation | last=Liu | first=Qing |title=Algebraic geometry and arithmetic curves| publisher=Oxford University Press | location=Oxford | isbn= 9780191547805 | MR=1917232 | year=2002}}

External links

{{springer |id=P/p075450|title=Proper morphism|author=V.I. Danilov}}
{{Citation | author1=The Stacks Project Authors | title=The Stacks Project | url=http://stacks.math.columbia.edu/}}

Category:Morphisms of schemes