---
created: 2022-02-10T23:42
updated: 2023-03-26T11:38
title: syntomic
aliases: [syntomic cohomology]
---

- Tags 
	- #arithmetic-geometry/prisms 
- Refs:
	- <https://arxiv.org/pdf/2202.04818.pdf>
- Links:
	- [etale cohomology](Unsorted/etale.md)
	- [Nygaard filtration](Nygaard%20filtration)
	- [smooth algebra](Unsorted/smooth%20algebra.md)
	- [perfectoid MOC](Unsorted/perfectoid%20MOC.md)
	- [etale comparison theorem](etale%20comparison%20theorem.md)
	- [Tate twist](Unsorted/l-adic%20cohomology.md)
	- [Unsorted/tor amplitude](Unsorted/tor%20amplitude.md)
	- [animated rings](animated%20rings.md)
	- [formal spectrum](Unsorted/formal%20spectrum.md)
	- [Hodge-Tate stack](Hodge-Tate%20stack)

---

# syntomic cohomology

Idea: finite presentation and locally complete intersection (fp+lci).


![](attachments/Pasted%20image%2020220210234222.png)
![](attachments/Pasted%20image%2020220210234907.png)
![](attachments/Pasted%20image%2020220323225333.png)
# the quasi-syntomic site

![](attachments/Pasted%20image%2020220210235134.png)


# Notes

The $(i, i)$ part of the syntomic complex should be related to [Milnor K theory](Milnor%20K%20theory).

![](attachments/Pasted%20image%2020220323225428.png)
![](attachments/Pasted%20image%2020220323225438.png)


# Spectral sequences

![](attachments/Pasted%20image%2020220323225644.png)
![](attachments/Pasted%20image%2020220323225656.png)
![](attachments/Pasted%20image%2020220323225702.png)