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date: 2022-04-09 23:12
modification date: Saturday 9th April 2022 23:12:11
title: "examples of K theory rings"
aliases: [examples of K theory rings]
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# Examples of K theory rings

- For $k$ a field, $\K_0(k) = \ZZ$.
- For $R$ a [Dedekind](Unsorted/Dedekind%20domain.md) domain with $K = \ff(R)$, we have $\K_0(R) \cong \ZZ \oplus \Cl(R)$, the [class group](Unsorted/class%20group.md).
- Letting $L$ be the class of the [tautological bundle](Unsorted/tautological%20bundle.md) and $1$ the trivial bundle, $$\K\left(\mathbb{C} P^{n}\right) \simeq \mathbb{Z}[L] /(1-L)^{n+1}$$
	- Alternatively description: let $L = \OO(-1)\dual = \OO(1)$ which has sections transverse to the zero section, so $e(L) = 1-t = [\CP^{n-1}]$, so define this class as a hyperplane $H$ to obtain $\K(\CP^n) = \ZZ[H]/H^{n+1}$.
- ![](attachments/Pasted%20image%2020220410185503.png)

- $\K_0(X) = \ZZ \oplus \Pic(X)$ for $X$ a curve.
- $\K_0(X) = \ZZ$ for $X$ a local Noetherian ring.
- $\KU^*(\CP^\infty) = \KU_*\fps{t}$ where $t = [\mcl] - [\ul{\CC}]$ with $\mcl\downto \BGL_1(\CC)$ the universal complex line bundle.
- $K_{i}\left(\mathbb{F}_{q}\right)=\left\{\begin{array}{l}\mathbb{Z} /\left(q^{n}-1\right) \text { if } i=2 n-1 \\ 0 \text { otherwise }\end{array}\right.$
	- So $\K_1(\FF) \cong \FF\units$ for $F$ a field.
- $\K(\FF_q)  = \bigoplus_{n\in \ZZ_{\geq 0}} \Sigma^{2n-1} C_{q^n-1}$
- $\K(\CC)$ is largely unknown
- $\K(\ZZ)$ is partially known.

## Categories

- $\K_0(\Fin\Set) = \ZZ$, $\K_1(\Fin\Set) =C_2$, and $\K(\Fin\Set) = \Omega^\infty \SS = QS^0$.

# Algebraic

![](attachments/Pasted%20image%2020221011184046.png)
![](attachments/Pasted%20image%2020221011184129.png)