---
created: 2022-03-17T21:11
updated: 2024-02-02T17:06
title: formal disk
aliases:
  - formal disc
  - formal disk
  - disk analogy
---


---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [valuation](Unsorted/Valuations.md)
	- [DVR](Unsorted/DVR.md)
	- [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md)
	- [Henselian](Unsorted/Hensel's%20Lemma.md)
	- [specialization](Unsorted/specialization.md)

---

# formal disk

- $\DD\mapstofrom \spec \CC[t] = \ts{(t), (0)}$ where $(t)$ is a closed point and $\eta \da (0)$ is an open generic point; 
	- $(t)$ corresponds to the origin, and $\eta$ is a neighbourhood of the origin
- Localize at $(0)$ to get $\CC\fls{t} = \ff \CC[t]$; thus $\DD^\circ \mapstofrom \spec \CC\fls{t}$.
- 

![](attachments/2023-03-31-55.png)
![](attachments/2023-03-31-56.png)

![](attachments/2023-03-31-57.png)

![](2023-03-31-85.png)

# Rings of integers

![Ring of integers of a nonarchimedean field](Unsorted/ring%20of%20integers#Ring%20of%20integers%20of%20a%20nonarchimedean%20field)

# Misc

Analogy:

![](attachments/Pasted%20image%2020220318195230.png)

![](attachments/Pasted%20image%2020220317211113.png)
![](attachments/Pasted%20image%2020220317211333.png)

![](attachments/Pasted%20image%2020220317211245.png)
![](attachments/Pasted%20image%2020220317211311.png)


# Milnor fibers
![](attachments/Pasted%20image%2020220317211437.png)
![](attachments/Pasted%20image%2020220317211557.png)

# Formal manifolds

![](2023-03-31-81.png)