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 $1+a_{1}z+a_2{z}^2+...\in A_{\rho }[[z]]$. 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.

Latex file using Diagrams.tex, 36 pages. To appear in "K-theory"