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

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

X × Y Z → Z {\displaystyle 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 [1]). One also says that X {\displaystyle X} is proper over Y {\displaystyle Y} . In particular, a variety X {\displaystyle X} over a field k {\displaystyle k} is said to be proper over k {\displaystyle k} if the morphism X → Spec ⁡ ( k ) {\displaystyle X\to \operatorname {Spec} (k)} is proper.

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 × k A 1 → A 1 {\displaystyle \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.

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:(EGA III, 3.2.4) let f : X → S {\displaystyle f\colon X\to S} be a morphism of finite type, S locally noetherian and F {\displaystyle F} a O X {\displaystyle {\mathcal {O}}_{X}} -module. If the support of F is proper over S, then for each i ≥ 0 {\displaystyle i\geq 0} the higher direct image R i f ∗ F {\displaystyle 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.^[10]
• 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.

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 x ¯ ∈ X ( R ) {\displaystyle {\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 x ¯ ∈ X ( R ) {\displaystyle {\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 x ¯ ∈ X ( R ) {\displaystyle {\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.

One of the motivating examples for the valuative criterion of properness is the interpretation of Spec ( C [ [ t ] ] ) {\displaystyle {\text{Spec}}(\mathbb {C} [[t]])} as an infinitesimal disk, or complex-analytically, as the disk Δ = { x ∈ C : | x | < 1 } {\displaystyle \Delta =\{x\in \mathbb {C} :|x|<1\}} . This comes from the fact that every power series

f ( t ) = ∑ n = 0 ∞ a n t n {\displaystyle f(t)=\sum _{n=0}^{\infty }a_{n}t^{n}}

converges in some disk of radius r {\displaystyle 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 {\displaystyle t} , this is the ring C [ [ t ] ] [ t − 1 ] = C ( ( t ) ) {\displaystyle \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 Δ ∗ = { x ∈ C : 0 < | x | < 1 } {\displaystyle \Delta ^{*}=\{x\in \mathbb {C} :0<|x|<1\}} with the origin removed. For a morphism of schemes over Spec ( C ) {\displaystyle {\text{Spec}}(\mathbb {C} )} , this is given by the commutative diagram

Δ ∗ → X ↓ ↓ Δ → Y {\displaystyle {\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 ∈ Δ {\displaystyle 0\in \Delta } in the image of Δ ∗ {\displaystyle \Delta ^{*}} .

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

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

C ( ( t ) ) ← C [ t , t − 1 ] ↑ ↑ C [ [ t ] ] ← C {\displaystyle {\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, Spec ( C [ [ t ] ] ) → Spec ( C [ t , t − 1 ] ) {\displaystyle {\text{Spec}}(\mathbb {C} [[t]])\to {\text{Spec}}(\mathbb {C} [t,t^{-1}])} , would imply there is a morphism C [ t , t − 1 ] → C [ [ t ] ] {\displaystyle \mathbb {C} [t,t^{-1}]\to \mathbb {C} [[t]]} sending t ↦ t {\displaystyle t\mapsto t} from the commutative diagram of algebras. This, of course, cannot happen. Therefore X {\displaystyle X} is not proper over Y {\displaystyle Y} .

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

C − { p } → X ↓ ↓ C → Y {\displaystyle {\begin{matrix}C-\{p\}&\rightarrow &X\\\downarrow &&\downarrow \\C&\rightarrow &Y\end{matrix}}}

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

Spec ( Frac ( O C , p ) ) → X ↓ ↓ Spec ( O C , p ) → Y {\displaystyle {\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 Spec ( Frac ( O C , p ) ) {\displaystyle {\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))} represents a local disk around p {\displaystyle {\mathfrak {p}}} with the closed point p {\displaystyle {\mathfrak {p}}} removed.

Let f : X → S {\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}} be a morphism between locally noetherian formal schemes. We say f is proper or X {\displaystyle {\mathfrak {X}}} is proper over S {\displaystyle {\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 → S 0 {\displaystyle f_{0}\colon X_{0}\to S_{0}} is proper, where X 0 = ( X , O X / I ) , S 0 = ( S , O S / K ) , I = f ∗ ( K ) O X {\displaystyle 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 S {\displaystyle {\mathfrak {S}}} .(EGA III, 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 g ^ : Y / Y 0 → Z / Z 0 {\displaystyle {\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 : X → S {\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}} be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on X {\displaystyle {\mathfrak {X}}} , then the higher direct images R i f ∗ F {\displaystyle R^{i}f_{*}F} are coherent.^[11]

• Proper base change theorem
• Stein factorization
• Tube lemma

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