---
date: 2022-03-23 13:15
modification date: Wednesday 23rd March 2022 13:15:50
title: formally smooth
aliases: [formally smooth, formal smoothness]

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


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [formally etale](Unsorted/formally%20etale.md)

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# formally smooth

Idea: $E \mapsvia{f} B$ is formally smooth if tangent vectors $\spec k[\eps]/\eps^2 \to B$ to tangent vectors $\spec k[\eps]/\eps^2 \to E$. 

Being [formally unramified](Unsorted/formally%20unramified.md) says there is at most one lift, and [formally etale](Unsorted/formally%20etale.md) means an isomorphism of tangent spaces.
Meant to look like a [submersion](submersion).

![](attachments/Pasted%20image%2020220407234042.png)

![](attachments/Pasted%20image%2020220802132052.png)