---
date: 2022-01-15 21:49
modification date: Monday 14th February 2022 09:25:32
title: torsor
aliases: [torsor, G torsor, G bundle, principal bundle, principal G bundle]

---

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


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [rational point](Unsorted/rational%20points.md)
	- [descent](Unsorted/descent.md)
	- [list of fibrations](Unsorted/list%20of%20fibrations.md)

---

# torsor

![](attachments/2023-02-24-torsor.png)

![](attachments/Pasted%20image%2020220501123055.png)

![](attachments/Pasted%20image%2020220421231848.png)


![](attachments/Pasted%20image%2020220420092018.png)
## Definitions
![](attachments/Pasted%20image%2020220319201751.png)
![](attachments/Pasted%20image%2020220319202154.png)


![](attachments/Pasted%20image%2020220214092540.png)
![G-torsors and principal bundles](attachments/Pasted%20image%2020210505014936.png)

![](attachments/Pasted%20image%2020220319201643.png)
-  A $G\dash$torsor is a set with a free transitive $G\dash$action.
For example, the fibers of a [principal bundle](principal bundle.md) are torsors.
Given any two torsors, we can compare them using elements of $G$, but there is no distinguished element.
For example, $\AA_n$ is a torsor over the vector space $k^n$.

![](attachments/Pasted%20image%2020220116001741.png)

As a [birational](Unsorted/birational.md) invariant and relation to being [unramified](Unsorted/unramified.md):
![](attachments/Pasted%20image%2020220319203715.png)

# Computation
![](attachments/Pasted%20image%2020220319203753.png)

# For elliptic curves
![](attachments/Pasted%20image%2020220214103055.png)

# In etale cohomology

![](attachments/Pasted%20image%2020220319203306.png)

# Relation to central simple algebras

![](attachments/Pasted%20image%2020220319203449.png)

# Examples

![](attachments/Pasted%20image%2020220421231918.png)

![](attachments/Pasted%20image%2020220319202851.png)
![](attachments/Pasted%20image%2020220319202909.png)

Something related to the [del Pezzo surface](del%20Pezzo%20surface) and [blowup](Unsorted/blowup.md):
![](attachments/Pasted%20image%2020220319203009.png)

Torsors of [algebraic groups](Unsorted/algebraic%20group.md):
![](attachments/Pasted%20image%2020220319203036.png)

For [elliptic curves](MOCs/elliptic%20curve.md):
![](attachments/Pasted%20image%2020220319203102.png)

Covering of affine curves:
![](attachments/Pasted%20image%2020220319203133.png)

Covering of varieties:
![](attachments/Pasted%20image%2020220319203150.png)

# Principal bundles

See [fiber bundle](Unsorted/fibration.md):
![](attachments/Pasted%20image%2020220403174007.png)