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

${\displaystyle X\times _{Y}Z\to Z}$

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 ${\displaystyle X}$ is proper over ${\displaystyle Y}$. In particular, a variety ${\displaystyle X}$ over a field ${\displaystyle k}$ is said to be proper over ${\displaystyle k}$ if the morphism ${\displaystyle X\to \operatorname {Spec} (k)}$ 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

${\displaystyle \mathbb {A} ^{1}\times _{k}\mathbb {A} ^{1}\to \mathbb {A} ^{1}}$

(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 ${\displaystyle f\colon X\to S}$ be a morphism of finite type, S locally noetherian and ${\displaystyle F}$ a ${\displaystyle {\mathcal {O}}_{X}}$-module. If the support of F is proper over S, then for each ${\displaystyle i\geq 0}$ the higher direct image ${\displaystyle R^{i}f_{*}F}$ 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 ${\displaystyle {\overline {x}}\in X(R)}$. (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 ${\displaystyle {\overline {x}}\in X(R)}$. (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 ${\displaystyle {\overline {x}}\in X(R)}$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Let ${\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}$ be a morphism between locally noetherian formal schemes. We say f is proper or ${\displaystyle {\mathfrak {X}}}$ is proper over ${\displaystyle {\mathfrak {S}}}$ if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map ${\displaystyle f_{0}\colon X_{0}\to S_{0}}$ is proper, where ${\displaystyle X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {S}},{\mathcal {O}}_{\mathfrak {S}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}}$ and K is the ideal of definition of ${\displaystyle {\mathfrak {S}}}$.(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 ${\displaystyle {\widehat {g}}\colon Y_{/Y_{0}}\to Z_{/Z_{0}}}$ on formal completions is a proper morphism of formal schemes.

Grothendieck proved the coherence theorem in this setting. Namely, let ${\displaystyle f\colon {\mathfrak {X}}\to {\mathfrak {S}}}$ be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on ${\displaystyle {\mathfrak {X}}}$, then the higher direct images ${\displaystyle R^{i}f_{*}F}$ are coherent.^[11]

• Proper base change theorem
• Stein factorization

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