---
date: 2022-01-15 21:49
modification date: Sunday 23rd January 2022 18:56:17
title: DVR
aliases: ["DVR", "DVRs", "discrete valuation"]

---

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

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- Tags: 
	- #NT/algebraic #CA
- Refs:
	- #todo/add-references
- Links:
	- [valuation](Unsorted/Valuations.md)
	- [dominant](Unsorted/dominant%20morphism.md)

---

# DVR

- Slogan:  a local PID that is not a field. The Noether regular local rings of dimension 1 are exactly the DVRs.
- Vague analogy: DVRs are supposed to look like $\DD \subseteq \CC$?
- Some nice properties of DVRs:
	- [Noetherian](Noetherian)
	- [PID](PID)
	- [local](Unsorted/localization%20of%20rings.md)
	- [Krull dimension](Unsorted/Krull%20dimension.md) one
	- [regular](Unsorted/regular%20ring.md)
	- [integrally closed](integrally%20closed)

![](attachments/Pasted%20image%2020220123185627.png)
![](attachments/Pasted%20image%2020220123185915.png)

- Scheme-theoretic properties: for $R\in\DVR$, $\spec R = \ts{ (0), \mfm}$ consists of a [generic point](generic%20point) and the [special point](special%20point).
- The ring of integers of a nonarchimedean local field is a complete DVR with finite residue field

![](attachments/Pasted%20image%2020220914181713.png)
![](attachments/Pasted%20image%2020220914181903.png)

![](attachments/Pasted%20image%2020221111232225.png)