The Whitehead group of the Novikov ring

Details The Whitehead group of the Novikov ring The Whitehead group of the Novikov ring

2000-12-05

arxiv.org/abs/math/0012031v1

K-Theory 21 (2000) 325-365

Mathematics

Algebraic Topology

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

Scientific paper

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.

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