The Whitehead group of the Novikov ring

Abstract

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