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- Tags: - #higher-algebra/K-theory - Refs: - #todo/add-references - Links: - Bott Periodicity - Hermitian K theory

complex K theory

Construction: take the categorical K-theory of \(({ \mathsf{Vect} }({\mathbf{C}}), \otimes_{\mathbf{C}})\). The associated FGL is the multiplicative formal group: \begin{align*} \begin{aligned} F_{{\operatorname{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} \end{align*}