----
> [!theorem] Theorem. ([[valuative criterion for properness]])
> ~
^theorem
Let $f:X\to Y$ be a [[scheme]] [[morphism of locally ringed spaces|morphism]] [[scheme morphism locally of finite type|of finite type]] with $X$ [[locally Noetherian scheme|Noetherian]]. Then the following are equivalent:
1. $f$ is [[proper scheme morphism|proper]];
2. For any [[field]] $k$ and [[valuation ring]] $R$ of $k$, given a commutative diagram
```tikz
\usepackage{tikz-cd}
\usepackage{amsmath}
\begin{document}
% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYAMpoYAYwAEAXwDWICaXSZc+QigCM5KrUYs2ADVnyQGbHgJEyqzfWatEHLr35DRkgEoGFJ5UXWXq1nTsATVlNGCgAc3giUAAzACcIAFskdRAcCCQAZn9tWxBYjwLElMQydMzEACY5OJKkcozU2uLkpCrqJsQsiQoJIA
\begin{tikzcd}
\text{Spec }k \arrow[d] \arrow[r] & X \arrow[d, "f"] \\
\text{Spec }R \arrow[r] & Y
\end{tikzcd}
\end{document}
```
there is a unique [[lifting|lift]][^1] $\text{Spec }R \to X$ making the diagram commute:
```tikz
\usepackage{tikz-cd}
\usepackage{amsmath}
\begin{document}
% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYAMpoYAYwAEAXwDWICaXSZc+QigCM5KrUYs2ADVnyQGbHgJEyqzfWatEHLr35DRkgEoGFJ5UXWXq1nTsATVlNGCgAc3giUAAzACcIAFskdRAcCCQAZn9tWxBYjwLElMQydMzEACY5OJKkcozU2uLkpCrqJsQsloS26s7KtIYsMHyoOjgAC3CQXJs2Tl4sOBw4AEJQiSA
\begin{tikzcd}
\text{Spec }k \arrow[d] \arrow[r] & X \arrow[d, "f"] \\
\text{Spec }R \arrow[r] \arrow[ru, "\exists!", dashed] & Y
\end{tikzcd}
\end{document}
```
> [!proof]- Proof. ([[valuative criterion for properness]])
> ~
----
####
[^1]: This part differs from the [[valuative criterion for separatedness]] by saying there *exists a unique* lift as opposed to there *exists at most one* lift.
-----
#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```