---
title: descent
aliases: [descent data, "faithfully flat descent", "Zariski descent", "etale descent"]
created: 2022-01-15T21:49
updated: 2023-06-05T18:53
---

---

- Tags 
	- #AG 
- Refs:
	- #todo/add-references
- Links:
	- [Barr-Beck](Unsorted/Barr-Beck.md)
	- [stacks MOC](Unsorted/stacks%20MOC.md)
	- [etale cohomology](Unsorted/l-adic%20cohomology.md)
	- [torsor](Unsorted/torsor.md)
	- [motive](Unsorted/motivic%20homotopy.md)
	- [etale morphism](Unsorted/etale.md)
	- [descent category](descent%20category.md)

---

# descent

![](2023-06-05.png)

![](attachments/Pasted%20image%2020220510160041.png)

## Ideas

![](attachments/Pasted%20image%2020220317190903.png)

![](attachments/Pasted%20image%2020220316123538.png)

## Definitions

## For modules
![Pasted image 20220115181849.png](attachments/Pasted%20image%2020220115181849.png)
![Pasted image 20220115182051.png](attachments/Pasted%20image%2020220115182051.png)

## For categories fibered in groupoids

![](attachments/Pasted%20image%2020220413092739.png)
![](attachments/Pasted%20image%2020220413092838.png)

![](attachments/Pasted%20image%2020220425234506.png)

## In HTT

![](attachments/Pasted%20image%2020220129012650.png)
![](attachments/Pasted%20image%2020220319001740.png)

## In terms of hypercovers

See also [[hyperdescent]].

![](attachments/Pasted%20image%2020220903171826.png)
![](attachments/Pasted%20image%2020220903172057.png)

# Misc


![](attachments/Pasted%20image%2020220220033741.png)
![](attachments/Pasted%20image%2020220317191921.png)

# Descent categories from monads
![](attachments/Pasted%20image%2020220317192127.png)
![](attachments/Pasted%20image%2020220317192213.png)
![](attachments/Pasted%20image%2020220317192317.png)

## Recovering descent for rings/modules

![](attachments/Pasted%20image%2020220317192435.png)
![](attachments/Pasted%20image%2020220317192448.png)
![](attachments/Pasted%20image%2020220317192538.png)
![](attachments/Pasted%20image%2020220317192605.png)
![](attachments/Pasted%20image%2020220317192719.png)
![](attachments/Pasted%20image%2020220317192816.png)
# Questions

- What is descent?
- What is [effective descent](effective%20descent)
- What is [Unsorted/descent](Unsorted/descent.md)?
- Types of descent:
	- What is [Zariski descent](Zariski%20descent.md)?
	- What is [Galois descent](Galois%20descent.md)?
	- What is [Unsorted/etale](Unsorted/etale.md)?
	- What is [fppf descent](fppf%20descent)?
	- What is [fpqc descent](fpqc%20descent)?
	- What is [descent](descent.md) for [quasicoherent sheaves](quasicoherent%20sheaf.md)?
- What is the connection to [Neron models](Neron%20models)? 
	See Bosch et al
- How is this related to [etale cohomology](Unsorted/etale%20cohomology.md)?





# Relation to torsors 

See [torsor](Unsorted/torsor.md) and [rational point](Unsorted/rational%20points.md).
For [elliptic curves](MOCs/elliptic%20curve.md), related to the [Selmer group](Unsorted/Selmer%20group.md)
![](attachments/Pasted%20image%2020220319204120.png)

![](attachments/Pasted%20image%2020220319204159.png)
![](attachments/Pasted%20image%2020220319204212.png)
![](attachments/Pasted%20image%2020220319204230.png)

# Applications

![](attachments/Pasted%20image%2020220404005629.png)