---
date: 2022-04-17 00:42
modification date: Sunday 17th April 2022 00:42:34
title: "etale morphism"
aliases: [etale morphism]
---

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- Refs:
	- #todo/add-references
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# etale morphism
![](attachments/Pasted%20image%2020220903173151.png)
![](attachments/Pasted%20image%2020220424193757.png)

An **etale morphism** $U\to X$ is:

- A [smooth](Unsorted/smooth%20scheme.md) (or [flat morphism](Unsorted/faithfully%20flat.md)) and an [unramified morphism](unramified%20morphism.md).
- A [smooth morphism](smooth%20morphism) of [relative dimension](relative%20dimension) zero.
- [formally etale](formally%20etale) and [locally of finite presentation](locally%20of%20finite%20presentation).
  
![](attachments/Pasted%20image%2020220418085754.png)

![](attachments/Pasted%20image%2020220420100914.png)

![](attachments/Pasted%20image%2020220417004236.png)
![](attachments/Pasted%20image%2020220417011932.png)

# finite etale morphism

![](attachments/Pasted%20image%2020220417005817.png)

![](attachments/Pasted%20image%2020220417014604.png)