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 → Y of schemes is called universally closed if for every scheme Z with a morphism Z → 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 ). 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. 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. 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 Yi and the restriction of f to all f−1(Yi) 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 XE is proper over E.
 * 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. 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.
 * 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. 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.
 * 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.
 * 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.
 * 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 $$\mathcal{O}_X$$-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.
 * 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 [x0,...,xn] 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 $$\mathbb{C}t[t^{-1}] = \mathbb{C}((t))$$ 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 $$\begin{matrix} \text{Spec}(\mathbb{C}((t))) & \to & \text{Spec}(\mathbb{C}[t,t^{-1}]) \\ \downarrow & & \downarrow \\ \text{Spec}(\mathbb{C}t) & \to & \text{Spec}(\mathbb{C}) \end{matrix}$$ where $$\text{Spec}(\mathbb{C}[t,t^{-1}]) = \mathbb{A}^1 - \{0\}$$ is the chart centered around $$\{x \}$$ on $$X$$. This gives the commutative diagram of commutative algebras $$\begin{matrix} \mathbb{C}((t)) & \leftarrow & \mathbb{C}[t,t^{-1}] \\ \uparrow & & \uparrow \\ \mathbb{C}t & \leftarrow & \mathbb{C} \end{matrix}$$ Then, a lifting of the diagram of schemes, $$\text{Spec}(\mathbb{C}t) \to \text{Spec}(\mathbb{C}[t,t^{-1}])$$, 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 $$\mathfrak{S}$$. 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.