$K_0$ of Burnside ring?

Question

I am trying to find $K_0(AG)$ for $G$ cyclic of prime order. According to a paper I have, tom Dieck and Petrie's paper "Geometric modules over the Burnside ring" (1978) shows $K_0(AG) = \mathbb{Z} \oplus \mathbb{Z}/((p-1)/2)$ for $p\geq 3$, but I cannot find this anywhere in the paper. Can somebody either explain where this is in the paper or point me to another source?

Answer 1

As $AG$ is noetherian of dimension 1*, finitely-generated projective $AG$-modules are classified by their rank and determinant, and so $$K_0(AG) = \mathbb{Z} \oplus Pic(AG).$$ Proposition 10.3.8 of tom Dieck--Petrie gives an exact sequence calculating $Pic(AG)$ in terms of the ghost map, and for $G=C_p$ cyclic of odd prime order this is easy to work out their sequence manually and get $$Pic(AC_p) \cong (\mathbb{Z}/p)^\times/\{\pm 1\} \cong \mathbb{Z}/((p-1)/2).$$

* Certainly $AG$ is finitely generated, so noetherian. As the ghost map $\Phi : AG \to CG \cong \mathbb{Z}^{\times N}$ is an injective ring homomorphism, and $\vert G \vert \cdot CG \subset \Phi(AG)$, it follows that $CG$ is a finite $AG$-module. But $CG$ is a product of 1-dimensional rings, so is 1-dimensional, and hence $AG$ is also 1-dimensional.

Addendum

For $G=C_p$ a quite easy direct calculation of $K_0(AC_p)$ is possible. There are only two transtive $C_p$-sets, $x := [C_p]$ and $1 := [C_p/C_p]$, and $C_p \times C_p$ is a free $C_p$-set of cardinality $p^2$, so represents $p[C_p]$. Thus as rings we have $$AC_p \cong \mathbb{Z}[x]/(x^2-px) =: R.$$ The homomorphism $f : R \to \mathbb{Z}$ sending $x$ to $p$ sends the ideal $(x)$ of $R$ isomorphically to the ideal $(p)$ of $\mathbb{Z}$, and $R/(x) \cong \mathbb{Z}$ so there is a Mayer--Vietoris sequence (see the K-book, III.2.6) $$K_1(\mathbb{Z}) \oplus K_1(\mathbb{Z}) \to K_1(\mathbb{Z}/p) \to K_0(R) \to K_0(\mathbb{Z}) \oplus K_0(\mathbb{Z}) \to K_0(\mathbb{Z}/p) \to 0.$$ Now the two maps $K_1(\mathbb{Z}) \to K_1(\mathbb{Z}/p)$ are equal and are $\mathbb{Z}^\times \to (\mathbb{Z}/p)^\times$ so the claimed calculation follows.