---
date: 2022-02-20 14:02
modification date: Sunday 20th February 2022 14:02:00
title: valuative criterion of properness
aliases: [valuative criterion of properness]

---

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

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# valuative criterion of properness

A criteria to check if a morphism of schemes is a [proper morphism](Unsorted/proper%20morphism.md).

Idea: for $R\in \DVR$ with $K = \ff(R)$, require 1-dimensional limits to exist.

- $\spec R\to Y$ is like a disc $\DD\subseteq Y$.
- $\spec K \to Y$ is like a punctured disc $\DD^\circ \subseteq Y$
- There should be one way to lift a disc $\DD\to X$ to $\DD^\circ \to Y$ and extend functions over the puncture.

Concretely, 

- $R = \spec \CC \fps{t}$ is a formal open disc (expansions of analytic functions at $z=0$ in $\CC$) and a DVR.
- Inverting $t$ yields $K = \CC\fls{t} = \ff(R)$ (formal Laurent series, expansions of meromorphic functions with poles only $z=0$.)
  
![](attachments/Pasted%20image%2020220622000635.png)


![](attachments/Pasted%20image%2020220220140319.png)


\begin{tikzcd}
	{\operatorname{Spec} \mathbb{C}((t)) \approx \mathbb{D}^\circ} && X \\
	\\
	{\operatorname{Spec} \mathbb{C}[[t]] \approx \mathbb{D}} && Y
	\arrow[from=3-1, to=3-3]
	\arrow["f", from=1-3, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow["{\exists !}"{description}, dashed, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG9wZXJhdG9ybmFtZXtTcGVjfSBcXG1hdGhiYntDfSgodCkpIFxcYXBwcm94IFxcbWF0aGJie0R9XlxcY2lyYyJdLFswLDIsIlxcb3BlcmF0b3JuYW1le1NwZWN9IFxcbWF0aGJie0N9W1t0XV0gXFxhcHByb3ggXFxtYXRoYmJ7RH0iXSxbMiwyLCJZIl0sWzIsMCwiWCJdLFsxLDJdLFszLDIsImYiXSxbMCwxXSxbMCwzXSxbMSwzLCJcXGV4aXN0cyAhIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)
> ![](attachments/Pasted%20image%2020220527230806.png)


![](attachments/Pasted%20image%2020220716090112.png)