The Whitehead group of the Novikov ring

Abstract

The Bass-Heller-Swan-Farrell-Hsiang-Siebenmann decomposition of the Whitehead group $K_1(A_ρ[z,z^{-1}])$ of a twisted Laurent polynomial extension $A_ρ[z,z^{-1}]$ of a ring $A$ is generalized to a decomposition of the Whitehead group $K_1(A_ρ((z)))$ of a twisted Novikov ring of power series $A_ρ((z))=A_ρ[[z]][z^{-1}]$. The decomposition involves a summand $W_1(A,ρ)$ which is an abelian quotient of the multiplicative group $W(A,ρ)$ of Witt vectors $1+a_1z+a_2z^2+... \in A_ρ[[z]]$. An example is constructed to show that in general the natural surjection $W(A,ρ)^{ab} \to W_1(A,ρ)$ is not an isomorphism.