Last modified date: <%+ tp.file.last_modified_date() %>
- Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links
Examples of K theory rings
-
For \(k\) a field, \({\mathsf{K}}_0(k) = {\mathbf{Z}}\).
-
For \(R\) a Dedekind domain with \(K = \operatorname{ff}(R)\), we have \({\mathsf{K}}_0(R) \cong {\mathbf{Z}}\oplus \operatorname{Cl} (R)\), the class group.
-
Letting \(L\) be the class of the tautological bundle and \(1\) the trivial bundle, \begin{align*}{\mathsf{K}}\left(\mathbb{C} P^{n}\right) \simeq \mathbb{Z}[L] /(1-L)^{n+1}\end{align*}
- Alternatively description: let \(L = {\mathcal{O}}(-1) {}^{ \vee }= {\mathcal{O}}(1)\) which has sections transverse to the zero section, so \(e(L) = 1-t = [{\mathbf{CP}}^{n-1}]\), so define this class as a hyperplane \(H\) to obtain \({\mathsf{K}}({\mathbf{CP}}^n) = {\mathbf{Z}}[H]/H^{n+1}\).
-
-
\({\mathsf{K}}_0(X) = {\mathbf{Z}}\oplus \operatorname{Pic}(X)\) for \(X\) a curve.
-
\({\mathsf{K}}_0(X) = {\mathbf{Z}}\) for \(X\) a local Noetherian ring.
-
\({\operatorname{KU}}^*({\mathbf{CP}}^\infty) = {\operatorname{KU}}_*{\left[\left[ t \right]\right] }\) where \(t = [{\mathcal{L}}] - [\underline{{\mathbf{C}}}]\) with \({\mathcal{L}}\downto \mathbf{B}\mkern-3mu \operatorname{GL} _1({\mathbf{C}})\) 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 \({\mathsf{K}}_1({ \mathbf{F} }) \cong { \mathbf{F} }^{\times}\) for \(F\) a field.
-
\({\mathsf{K}}({ \mathbf{F} }_q) = \bigoplus_{n\in {\mathbf{Z}}_{\geq 0}} \Sigma^{2n-1} C_{q^n-1}\)
-
\({\mathsf{K}}({\mathbf{C}})\) is largely unknown
-
\({\mathsf{K}}({\mathbf{Z}})\) is partially known.
Categories
- \({\mathsf{K}}_0({\mathsf{Fin}}{\mathsf{Set}}) = {\mathbf{Z}}\), \({\mathsf{K}}_1({\mathsf{Fin}}{\mathsf{Set}}) =C_2\), and \({\mathsf{K}}({\mathsf{Fin}}{\mathsf{Set}}) = \Omega^\infty {\mathbb{S}}= QS^0\).
Algebraic
