---
created: 2022-04-05T23:42
updated: 2024-01-23T11:49
aliases:
  - prismatic
  - prism
  - Prism
  - prismatic cohomology
---

- Tags 
	- #arithmetic-geometry/p-adic-hodge-theory  
- References:
	- <http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/lecture1-overview.pdf> #resources/notes 
	- <http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/> #resources/notes 
	- Yuri's giant reading list:
	<https://www.math.brown.edu/ysulyma/reading.html> #resources/recommendations 
	- Bhatt-Lurie 2022, Absolute Prismatic Cohomology. <https://arxiv.org/pdf/2201.06120.pdf#page=1> #resources/papers/2022

- Links:
	- [Unsorted/Masterclass in Condensed Mathematics](Unsorted/Masterclass%20in%20Condensed%20Mathematics.md)
	- The [Nygaard filtration](Nygaard%20filtration.md)
	- [Tate twist](Tate%20twist)
	- [perfectoid MOC](Unsorted/perfectoid%20MOC.md)
	- [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md)
	- [spec Z as a curve](spec%20Z%20as%20a%20curve.md)
	- [topological K theory](topological%20K%20theory)
	- [MMP](Unsorted/minimal%20model%20program.md)
	- [p-adic etale K theory](Unsorted/topological%20K%20theory.md)
	- [Riemann-Hilbert correspondence](Unsorted/Riemann-Hilbert%20correspondence.md)

# prismatic cohomology

## Prisms

![](attachments/2023-03-05prisms.png)

See [delta rings](delta%20rings)

# Motivations

![](attachments/Pasted%20image%2020220515001317.png)

A cohomology theory for [mixed characteristic](Unsorted/mixed%20characteristic.md) rings, inspired by calculations in [stable homotopy](Unsorted/stable%20homotopy.md) and [Galois representations](Unsorted/Galois%20representations.md).
Used in the proof of a [mixed characteristic](Unsorted/mixed%20characteristic.md) analog of [Kodaira vanishing](Unsorted/Kodaira%20vanishing.md), yielding a [minimal model program](Unsorted/minimal%20model%20program.md) in the [birational geometry](Unsorted/birational%20geometry.md) of [arithmetic threefolds](arithmetic%20threefolds).
Used in a proof of the [Bott vanishing theorem](Bott%20vanishing%20theorem) in [algebraic K theory](Unsorted/K-theory.md).

![](attachments/Pasted%20image%2020220515020039.png)
![](attachments/Pasted%20image%2020220515020106.png)
![](attachments/Pasted%20image%2020220515020128.png)

![](attachments/Pasted%20image%2020220515001924.png)
![](attachments/Pasted%20image%2020220515001930.png)

Motivations from [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md), [THH](Unsorted/THH.md)
![](attachments/Pasted%20image%2020220515002005.png)

![](attachments/Pasted%20image%2020220515002117.png)

# Definitions

![](attachments/Pasted%20image%2020220323160918.png)
![](attachments/Pasted%20image%2020220323161013.png)
![](attachments/Pasted%20image%2020220323165142.png)
![](attachments/Pasted%20image%2020220323165223.png)
![](attachments/Pasted%20image%2020220323165235.png)
# Examples

![](attachments/Pasted%20image%2020220323160945.png)


# Notes



![](attachments/Pasted%20image%2020220128233018.png)

![](attachments/Pasted%20image%2020220128233155.png)
![](attachments/Pasted%20image%2020220128233242.png)

# Syntomic complexes

See [Tate twist](Unsorted/l-adic%20cohomology.md), [motivic cohomology](Unsorted/motivic%20cohomology.md).
![](attachments/Pasted%20image%2020220323170213.png)

# Applications

![](attachments/Pasted%20image%2020220515002207.png)