---
date: 2022-03-25 20:56
modification date: Friday 25th March 2022 20:56:42
title: "Morava K theory"
aliases: [Morava K theory]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #homotopy/stable-homotopy/equivariant #higher-algebra/K-theory 
- Refs:
	- <https://etale.site/writing/ustars-tmf-beamer.pdf#page=103> #resources/slides #resources/summaries 
- Links:
	- [algebraic K theory](Unsorted/K-theory.md)
	- [chromatic homotopy](Unsorted/chromatic%20homotopy%20theory.md)
	- [moduli stack of formal groups](Unsorted/moduli%20of%20formal%20group%20laws.md)
	- [local spectra](local%20spectra.md)

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# Morava K theory

![](attachments/Pasted%20image%2020220508204301.png)
![](attachments/Pasted%20image%2020220508204422.png)

# Motivations

#why-care: these represent cohomology theories with Kunneth isomorphisms (not just a Kunneth SES). ![](attachments/Pasted%20image%2020220508204157.png)


![](attachments/Pasted%20image%2020220419175018.png)
![](attachments/Pasted%20image%2020220419175225.png)

![](attachments/Pasted%20image%2020220325224815.png)

![](attachments/Pasted%20image%2020220325205653.png)
![](attachments/Pasted%20image%2020220325205742.png)

This last property can be interpreted as the statement that the [Tate fixed point](Tate%20fixed%20point) spectrum $K(n)^{t G}$ is trivial for any $G\in\Fin\Grp$.

![](attachments/Pasted%20image%2020220325205912.png)
![](attachments/Pasted%20image%2020220325230139.png)
![](attachments/Pasted%20image%2020220325230151.png)
![](attachments/Pasted%20image%2020220325230208.png)

See [split epi](split%20epi).

# Construction

![](attachments/Pasted%20image%2020220325232659.png)
![](attachments/Pasted%20image%2020220325232722.png)