---
date: 2022-04-05 23:42
modification date: Monday 18th April 2022 12:17:47
title: "proper morphism"
aliases: [proper]
---

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

---
- Tags: 
	- #AG/basics 
- Refs:
	- #todo/add-references
- Links:
	- [flat morphism](Unsorted/flat%20morphism.md)
	- [projective morphism](projective%20morphism)
	- [separated morphism](Unsorted/separated.md)

---

# proper morphism

```ad-summary
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)
```

![](attachments/Pasted%20image%2020220914155542.png)
![](attachments/Pasted%20image%2020220914155611.png)

A morphism $X\mapsvia f Y$ 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.

![](attachments/Pasted%20image%2020220914155841.png)

![](attachments/Pasted%20image%2020220914160439.png)

## 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](Unsorted/universally%20closed.md), [separated](Unsorted/separated.md), [of finite type](of%20finite%20type).
	- 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 $X\in \Var\slice k$ is **proper** iff the structure morphism $X\to \spec k$ is a proper morphism.

# Checking properness

In terms of the [valuative criterion of properness](Unsorted/valuative%20criterion%20of%20properness.md), for $f:X\to Y$ a morphism [of finite type](of%20finite%20type) of [Noetherian schemes](Unsorted/Noetherian%20scheme.md), given a [regular](Unsorted/regular%20scheme.md) curve $C$ on $Y$ corresponding to $\spec R\to Y$ (for $R$ a [[DVR]]) and a lift of the [generic point](generic%20point) of $C$ to $X$, there is exactly one way to complete the curve with lifts of [closed points](Unsorted/special%20fiber.md).

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](Unsorted/toric.md) induced by $\Phi: N \rightarrow N^{\prime}$. 
TFAE:

(1) $\phi$ is proper as a continuous map of topological spaces.
(2) $\phi$ is proper as a morphism of algebraic varieties.
(3) $\bar{\Phi}_{\mathbb{R}}^{-1}\left(\left|\Sigma^{\prime}\right|\right)=|\Sigma|$.

# Examples

- For $\AA^1\times \PP^1$, $\proj_1$ is proper and $\proj_2$ is not proper.
- Continuous maps from a compact space to a Hausdorff space.
- A scheme $X\in \Sch^\ft\slice \CC$ is proper iff $X(\CC)$ is compact Hausdorff.
- Any [closed immersion](Unsorted/closed%20immersion.md) 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](finite%20morphism) (in fact, a morphism is finite iff proper and [quasi-finite](Unsorted/locally%20quasi-finite.md)).
- Any [projective morphism](projective%20morphism).
- For any ring $R$, projective space $\PP^n\slice R\to \spec R$.
- Any $X\in \Aff\Var\slice k$ with $\dim X \geq 1$ is **never** proper over $k$.
- $\AA^1\slice k$ is never proper since $\AA^1\slice k\to \spec k$ is not universally closed: check  
  $$\AA^1\fiberprod{\spec k} \AA^1\mapsvia{(x,y)\mapsto y} \AA^1.$$
- $\AA^1\slice \CC\to \spec \CC$ is not proper, since $\CC$ is not compact.
  But it is a separated finite type closed morphism, necessitating [universally closed](Unsorted/universally%20closed.md).
  
# Exercises

![](attachments/Pasted%20image%2020220516191823.png)