---
date: 2022-04-19 17:12
modification date: Tuesday 19th April 2022 17:12:13
title: "complex K theory"
aliases: [complex K theory]
---

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

---
- Tags: 
	- #higher-algebra/K-theory 
- Refs:
	- #todo/add-references
- Links:
	- [Bott Periodicity](Unsorted/Bott%20Periodicity.md)
	- [Hermitian K theory](Hermitian%20K%20theory)

---

# complex K theory


 Construction: take the categorical [K-theory](Unsorted/K-theory.md) of $(\Vect(\CC), \tensor_\CC)$.
 The associated [FGL](Unsorted/Formal%20group.md) is the multiplicative formal group:
 $$
 \begin{aligned}
F_{\KU}(x, y) &=\left[\mathscr{L}_{1} \otimes \mathscr{L}_{2}\right]-1 \\
&=\left[\mathscr{L}_{1}\right] \cdot\left[\mathscr{L}_{2}\right]-1 \\
&=\left(\left[\mathscr{L}_{1}\right] \cdot\left[\mathscr{L}_{2}\right]-\left[\mathscr{L}_{1}\right]-\left[\mathscr{L}_{2}\right]+1\right)+\left[\mathscr{L}_{1}\right]-1+\left[\mathscr{L}_{2}\right]-1 \\
&=\left(\left[\mathscr{L}_{1}\right]-1\right) \cdot\left(\left[\mathscr{L}_{2}\right]-1\right)+\left(\left[\mathscr{L}_{1}\right]-1\right)+\left(\left[\mathscr{L}_{2}\right]-1\right) \\
&=x y+x+y .
\end{aligned}
$$

![](attachments/Pasted%20image%2020220422152729.png)

![](attachments/Pasted%20image%2020220422153515.png)