---
date: 2022-01-15 21:49
modification date: Friday 11th February 2022 12:38:17
title: Algebraic de Rham cohomology
aliases: [algebraic de Rham complex, algebraic de Rham cohomology, "de Rham cohomology", "de Rham", "algebraic de Rham", de Rham complex, Kahler differentials, relative differentials, algebraic differentials, log de Rham]

---


---

- Tags 
	- #AG #arithmetic-geometry 
- Refs:
	- <https://link.springer.com/content/pdf/10.1007%2F978-3-319-50926-6_3.pdf> #resources/papers
- Links:
	- [resolution of singularities](Unsorted/resolution%20of%20singularities.md)
	- [differential forms](Unsorted/differential%20forms.md)
	- For connections:
		- [Unsorted/curvature form](Unsorted/curvature%20form.md)
		-  [Unsorted/torsion of a connection](Unsorted/torsion%20of%20a%20connection.md)
	-  [Unsorted/de Rham isomorphism](Unsorted/de%20Rham%20isomorphism.md)

---

# algebraic de Rham cohomology


Closed vs exact:
![](attachments/Pasted%20image%2020220202222055.png)

# Definitions

![](attachments/Pasted%20image%2020220502142639.png)
![](attachments/Pasted%20image%2020220502142723.png)


![](attachments/Pasted%20image%2020220315102609.png)

![](attachments/Pasted%20image%2020220209092756.png)
![](attachments/Pasted%20image%2020220209092911.png)
![](attachments/Pasted%20image%2020220209093411.png)

## Log de Rham

![](attachments/2023-03-08log.png)

# Properties

![](attachments/Pasted%20image%2020220502142654.png)

# Degeneration of the spectral sequence

![](attachments/Pasted%20image%2020220502142804.png)

# Logarithmic differentials
![](attachments/Pasted%20image%2020220209094009.png)
![](attachments/Pasted%20image%2020220209094224.png)
Comparison to [Crystalline cohomology](Crystalline%20cohomology) of the reduction mod $p$:
![](attachments/Pasted%20image%2020220209094344.png)
![](attachments/Pasted%20image%2020220209094318.png)
Frobenius action and [Grothendieck-Lefschetz Trace Formula](Unsorted/Grothendieck-Lefschetz%20Trace%20Formula.md):
![](attachments/Pasted%20image%2020220209094418.png)
# Examples

![](attachments/Pasted%20image%2020220505161230.png)

![](attachments/Pasted%20image%2020220128233100.png)

# Grothendieck's Proof
![](attachments/Pasted%20image%2020220211123817.png)
![](attachments/Pasted%20image%2020220211123951.png)

# Derivations
![](attachments/Pasted%20image%2020220213201126.png)

# Motivations

From physics:
![](attachments/Pasted%20image%2020220213222929.png)

# Exercises

- Affine space:
![](attachments/Pasted%20image%2020220202222035.png)

- Elliptic curves:
![](attachments/Pasted%20image%2020220202222148.png)
![](attachments/Pasted%20image%2020220202222201.png)