---
date: 2022-02-07 12:10
modification date: Monday 7th February 2022 12:10:53
title: l-adic cohomology
aliases: [l-adic, Tate twist, etale cohomology]

---


---

- Tags 
	- #arithmetic-geometry 
- Refs:
	- <https://people.maths.ox.ac.uk/brantner/p-adic.pdf> #resources/notes
	- [Milne's course notes](https://www.jmilne.org/math/CourseNotes/LEC.pdf) #resources/course-notes 
	- <https://www.youtube.com/embed/P8SqRrnOs20>
	- Course notes: <https://www.dropbox.com/s/nn6p6wp94tlcj77/Etale.pdf?dl=0> #resources/course-notes 
	- <http://math.colorado.edu/~jonathan.wise/teaching/math8174-spring-2014/notes.pdf#page=56> #resources/course-notes 
- Links:
	- [Weil Conjectures](Unsorted/Weil%20Conjectures.md) 
	- [Kummer sequence](Kummer%20sequence.md)
	- [direct image](direct%20image.md)
	- [etale cohomology](Unsorted/l-adic%20cohomology.md)
	- [constructible sheaf](constructible%20sheaf)
	- [Tsen's theorem](Unsorted/Tsen's%20theorem.md)
	- [Leray spectral sequence](Leray%20spectral%20sequence)
	- [Zariski's main theorem](Unsorted/Zariski's%20main%20theorem.md)
	- [Tate twist](Unsorted/l-adic%20cohomology.md)
	- [Weil cohomology](Unsorted/Weil%20cohomology.md)
	- [subcanonical](subcanonical.md)

---

# Tate twist

![](attachments/Pasted%20image%2020220726153830.png)

![](attachments/Pasted%20image%2020220503105033.png)

# l-adic cohomology

![](attachments/Pasted%20image%2020220318193105.png)
![](attachments/Pasted%20image%2020220318193258.png)

![](attachments/Pasted%20image%2020220207121019.png)

![](attachments/Pasted%20image%2020220207121119.png)

Poincare duality:
![](attachments/Pasted%20image%2020220207121209.png)
![](attachments/Pasted%20image%2020220207121245.png)
![](attachments/Pasted%20image%2020220207121356.png)

See also [weak and hard Lefschetz theorems](weak%20and%20hard%20Lefschetz%20theorems)

![](attachments/Pasted%20image%2020220317211414.png)

# Properties

![](attachments/Pasted%20image%2020220423001012.png)
![](attachments/Pasted%20image%2020220423001021.png)
![](attachments/Pasted%20image%2020220423001212.png)
![](attachments/Pasted%20image%2020220423001338.png)

# Etale cohomology

![](attachments/Pasted%20image%2020220414211728.png)
![](attachments/Pasted%20image%2020220414212015.png)

See [examples of computations of etale cohomology](Unsorted/examples%20of%20computations%20of%20etale%20cohomology.md)
![](attachments/Pasted%20image%2020220404003555.png)
![](attachments/Pasted%20image%2020220404003623.png)
![](attachments/Pasted%20image%2020220404010433.png)

## Poincare duality

![](attachments/Pasted%20image%2020220404004458.png)

## Exact sequences
![](attachments/Pasted%20image%2020220404010502.png)

A [spectral sequence](Unsorted/spectral%20sequence.md) converging to [algebraic K theory](Unsorted/K-theory.md).
![](attachments/Pasted%20image%2020220404010913.png)

## Excision
![](attachments/Pasted%20image%2020220404010630.png)

# amitsur complex
![](attachments/Pasted%20image%2020220209190907.png)

# Comparison

![](attachments/Pasted%20image%2020220503104950.png)

![](attachments/Pasted%20image%2020221231193740.png)
![](attachments/Pasted%20image%2020221231193809.png)