---
date: 2022-03-18 19:34
modification date: Friday 18th March 2022 19:34:02
title: crystalline cohomology
aliases: [crystalline, divided power thickening]

---


---

- Tags 
	- #arithmetic-geometry #arithmetic-geometry/p-adic-hodge-theory 
- Refs:
	- <https://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/hokudai.pdf> #resources/notes 
	- <http://www-personal.umich.edu/~malloryd/haoyang.pdf>
	- Reading seminar: <https://math.berkeley.edu/~ogus/Seminar%20on%20BLM/blmindex.html>
- Links:
	- [Galois representations](Unsorted/Galois%20representations.md)
	- [arithmetic geometry MOC](Unsorted/arithmetic%20geometry%20MOC.md)
	- [divided power](divided%20power)
	- [semilinear action](semilinear%20action.md)
	- [formal disk](Unsorted/formal%20disk.md)
	- [period ring](Unsorted/period.md)
	- [admissible representation](Unsorted/admissible%20representation.md)
	- [inertia group](Unsorted/ramification%20index.md)
	- [cyclotomic character](cyclotomic%20character)
	- [perfect ring](perfect%20ring.md)
	- [Galois descent](Galois%20descent.md)
	- [Tate twist](Unsorted/l-adic%20cohomology.md)
	- [de Rham-Witt complex](Unsorted/de%20Rham-Witt.md)
	- [etale comparison theorem](Unsorted/etale%20comparison%20theorem.md)
	- Types of representations:
		- [crystalline representation](crystalline%20representation.md)
		- [semistable represenation](semistable%20represenation)
		- [potentially semistable representation](potentially%20semistable%20representation)
		- [de Rham representation](de%20Rham%20representation)
		- [Hodge-Tate representation](Hodge-Tate%20representation.md)
	- See [p-adic monodromy theorem](p-adic%20monodromy%20theorem.md)
	- [de Rham crystalline comparison](Unsorted/de%20Rham%20crystalline%20comparison.md)
	- [crystalline representation](crystalline%20representation.md)

---

# crystalline cohomology

![](attachments/2023-02-15-defcrys.png)

# Motivation

![](attachments/Pasted%20image%2020220514181006.png)

![](attachments/Pasted%20image%2020220502150933.png)

![](attachments/Pasted%20image%2020220410155415.png)

![](attachments/Pasted%20image%2020220318193424.png)

Relation to [Weil cohomology](Unsorted/Weil%20cohomology.md) for [smooth](Unsorted/smooth%20scheme.md) and [proper schemes](proper%20schemes), [algebraic de Rham cohomology](Unsorted/algebraic%20de%20Rham%20cohomology.md), and [Witt vectors](Archive/AWS2019/Witt%20vectors.md)
![](attachments/Pasted%20image%2020220318193616.png)

![](attachments/Pasted%20image%2020220502145248.png)
# Definition

![](attachments/Pasted%20image%2020220502184129.png)
![](attachments/Pasted%20image%2020220502184150.png)

![](attachments/Pasted%20image%2020220318194718.png)
![](attachments/Pasted%20image%2020220318194842.png)

See [closed immersion](Unsorted/closed%20immersion.md), [special fiber](Unsorted/special%20fiber.md), [generic fiber](generic%20fiber), [good reduction](Unsorted/good%20reduction.md), [semistable reduction](semistable%20reduction).

# Frobenius

![](attachments/Pasted%20image%2020220318195025.png)


# Comparisons

Several naturally occurring varieties in number theory do not possess such a well-behaved reduction, a famous example being the [Tate curve](Tate%20curve). So replace with [semistable reduction](semistable%20reduction).

![](attachments/Pasted%20image%2020220318195311.png)
![](attachments/Pasted%20image%2020220318195814.png)

## Periods

![](attachments/Pasted%20image%2020220318201405.png)
One can recover real de Rham cohomology by taking fixed points on the right hand side.

![](attachments/Pasted%20image%2020220318201506.png)

## Algebraic de Rham to etale

![](attachments/Pasted%20image%2020220318203927.png)


See [B_dr](B_dr.md)

## Etale to crystalline
![](attachments/Pasted%20image%2020220318204001.png)
![](attachments/Pasted%20image%2020220318204024.png)
![](attachments/Pasted%20image%2020220318204110.png)

See [Dieudonne module](Dieudonne%20module), [abelian variety](abelian%20variety), [Tate module](Unsorted/Tate%20module.md), [Galois representations](Unsorted/Galois%20representations.md), [mysterious functor](mysterious%20functor), [generic fiber](generic%20fiber), [Hodge filtration](Hodge%20filtration).

## Etale to log crystalline

![](attachments/Pasted%20image%2020220318195531.png)
![](attachments/Pasted%20image%2020220318195554.png)

![](attachments/Pasted%20image%2020220318195650.png)

See [monodromy operator](monodromy%20operator),
![](attachments/Pasted%20image%2020220318195735.png)

![](attachments/Pasted%20image%2020220318205316.png)

# Notes

Relation to [admissible representation](Unsorted/admissible%20representation.md):
![](attachments/Pasted%20image%2020220318210100.png)

Use of [Hilbert 90](Unsorted/Hilbert%2090.md) and [Faltings theorem](Unsorted/Faltings%20theorem.md):
![](attachments/Pasted%20image%2020220318210138.png)
![](attachments/Pasted%20image%2020220318210157.png)