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

Abstract

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 elementwhere the period p = p(m) is: 2(ℓ − 1)ℓ ν−1 if m = ℓ ν and ℓ is an odd prime; max{8, 2 ν−1 } if m = 2 ν ; and p(m i ) if m = m i 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; 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 F with 1/m ∈ R. Then the Bott periodicity mapsTheorem 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 b K : The oldest result of this kind is due to Browder, who proved in [1, 2.6] that the Bott periodicity map b K 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 mapProof. If m is the nilradical of B, then B red = B/m is semisimple. As such it is a product of matrix rings over finite fields. Now K * (B; Z/m) ∼ = K * (B red ; 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 K n (F 2 [C 2 ]; Z/8) were computed by Hesselholt and Madsen in [4]; they are not Bott periodic as their order goes to infinity with n.