Bott Periodicity for group rings An Appendix to “Periodicity of Hermitian K-groups”

Abstract

We show that the groups Kn(RG; Z/m) are Bott-periodic for n≥1 whenever G is a finite group, m is prime to |G|, R is a ring of S-integers in a number field and 1/m ∈ R. For any positive integer m there is a Bott element bK ∈ Kp(Z[1/m]; Z/m), where the period p = p(m) is: 2(l−1)l if m = l and l is odd; max{8, 2} if m = 2; and ∏ p(mi) if m = ∏ mi is the factorization of m into primary factors. In this appendix we consider a finite group G of order prime to m, and consider the Bott periodicity map x 7→ x · bK from Kn(R[G]; Z/m) to Kn+p(R[G]; Z/m) for rings of integers R in local and global fields. Theorem 0.1. Assume that m is relatively prime to |G|, and that R is a ring of S-integers in a number field with 1/m ∈ R. Then the Bott periodicity maps bK : Kn(R[G]; Z/m) → Kn+p(R[G]; Z/m) are isomorphisms for all n ≥ 1. Theorem 0.2. Assume that m is relatively prime to |G|, and that R is the ring of integers in a local field F with 1/m ∈ R. Then the Bott periodicity maps bK : Kn(R[G]; Z/m) → Kn+p(R[G]; Z/m) are isomorphisms for all n ≥ 0, Theorems 0.1 and 0.2 are used in [BKO] to show that KQn(R[G]; Z/m) also satisfies Bott periodicity. Since this is immediate if m is odd, when the non-Witt part of KQn(A; Z/m) is a summand of Kn(A; Z/m), this result is primarily interesting for m even and G a (solvable) group of odd order. Remark. The proofs show that we may replace R[G] by any order in F [G]. The oldest result of this kind is due to Browder, who proved in [1, 2.6] that the Bott periodicity map bK is an isomorphism for finite fields and n ≥ 0 (when m is prime, which implies periodicity for all m). Almost as old is the following folklore result, which includes finite group rings. Lemma 0.3. If B is a finite ring and 1/m ∈ B, the the Bott periodicity map Kn(B, Z/m) → Kn+p(B, Z/m) is an isomorphism for all n ≥ 0. Proof. If m is the nilradical of B, then Bred = B/m is semisimple. As such it is a product of matrix rings over finite fields. Now K∗(B; Z/m) ∼= K∗(Bred; Z/m) by [9, 1.4]. By Morita invariance, we are reduced to the Browder’s theorem that the Bott periodicity map is an isomorphism for finite fields. Remark 0.4. The finite groups Kn(F2[C2]; Z/8) were computed by Hesselholt and Madsen in [4]; they are not Bott periodic as their order goes to infinity with n. The next step is to consider the semisimple group ring F [G] when F is a number field. We will use the fact that multiplication by bK is an isomorphism on