Last modified date: <%+ tp.file.last_modified_date() %>

- Tags: - #AG/basics - Refs: - #todo/add-references - Links: - flat morphism - projective morphism - separated morphism

proper morphism

Proper variety: **analagous to a closed compact complex manifold.**
Proper morphism: **closed with compact fibers**.
Valuative criteria of properness: lifts of curves into $X$ can be extended to a **compact** curves in $X$.

The most important properties:
![](attachments/Pasted%20image%2020220914160609.png)

A morphism \(X\xrightarrow{f} Y\) is proper when for a curve \(C \to Y\) with a lift of the generic point to \(X\), the rest of the curve can be lifted uniquely.

Definition

• A continuous map of topological spaces \(f: X \rightarrow Y\) is proper if \(f^{-1}(T)\) is compact for all compact \(T \subseteq Y\), so preimages of compact sets are compact.
  • Alternatively, \(f\) is proper iff \(f\) is a continuous closed map (so images of closed sets are closed) with compact fibers. Equivalent to the above definition if the target space is Hausdorff and locally compact.
• A morphism of schemes is proper if it is universally closed, separated, of finite type.
  • A morphism of algebraic varieties \(\phi: X \rightarrow Y\) is proper if it is universally closed, i.e., if for all \(\psi: Z \rightarrow Y\), the mapping \(\phi^{\prime}: X \times \times_{Y} Z \rightarrow Z\) is a closed mapping in the Zariski topology.
• A variety $X\in {\mathsf{Var}}_{/ {k}} $ is proper iff the structure morphism \(X\to \operatorname{Spec}k\) is a proper morphism.

Checking properness

In terms of the valuative criterion of properness, for \(f:X\to Y\) a morphism of finite type of Noetherian schemes, given a regular curve \(C\) on \(Y\) corresponding to \(\operatorname{Spec}R\to Y\) (for \(R\) a DVR) and a lift of the generic point of \(C\) to \(X\), there is exactly one way to complete the curve with lifts of closed points.

Necessary condition: all fibers are proper. However, this isn’t sufficient - for example, consider an open embedding.

For toric varieties

Let \(\phi: X_{\Sigma} \rightarrow X_{\Sigma}^{\prime}\) be a toric morphism of normal toric varieties induced by \(\Phi: N \rightarrow N^{\prime}\). TFAE:

• \(\phi\) is proper as a continuous map of topological spaces.
• \(\phi\) is proper as a morphism of algebraic varieties.
• \(\mkern 1.5mu\overline{\mkern-1.5mu\Phi\mkern-1.5mu}\mkern 1.5mu_{\mathbb{R}}^{-1}\left(\left|\Sigma^{\prime}\right|\right)=|\Sigma|\).

Examples

• For \({\mathbf{A}}^1\times {\mathbf{P}}^1\), \(\mathop{\mathrm{proj}}_1\) is proper and \(\mathop{\mathrm{proj}}_2\) is not proper.
• Continuous maps from a compact space to a Hausdorff space.
• A scheme $X\in {\mathsf{Sch}}^{\mathrm{ft}}_{/ {{\mathbf{C}}}} $ is proper iff \(X({\mathbf{C}})\) is compact Hausdorff.
• Any closed immersion is proper – they are separated since any affine morphism is, finite type, and closed immersions are preserved under base change, so universally closed.
• Any finite morphism (in fact, a morphism is finite iff proper and quasi-finite).
• Any projective morphism.
• For any ring \(R\), projective space \({\mathbf{P}}^n_{/ {R}} \to \operatorname{Spec}R\).
• Any $X\in {\mathsf{Aff}}{\mathsf{Var}}_{/ {k}} $ with \(\dim X \geq 1\) is never proper over \(k\).
• ${\mathbf{A}}^1_{/ {k}} $ is never proper since \({\mathbf{A}}^1_{/ {k}} \to \operatorname{Spec}k\) is not universally closed: check
  \begin{align*}{\mathbf{A}}^1 \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} {\mathbf{A}}^1\xrightarrow{(x,y)\mapsto y} {\mathbf{A}}^1.\end{align*}
• \({\mathbf{A}}^1_{/ {{\mathbf{C}}}} \to \operatorname{Spec}{\mathbf{C}}\) is not proper, since \({\mathbf{C}}\) is not compact. But it is a separated finite type closed morphism, necessitating universally closed.

Exercises