Commentarii Mathematici Helvetici

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Volume 88, Issue 3, 2013, pp. 613–642
DOI: 10.4171/CMH/298

Published online: 2013-08-13

A splitting for K1 of completed group rings

Peter Schneider[1] and Otmar Venjakob[2]

(1) Universität Münster, Germany
(2) Universität Heidelberg, Germany

For $p\neq 2$ and a uniform pro-$p$ group $G$ and its Iwasawa algebras $\Lambda (G) := \mathbb{Z}_{p}[\hskip-.7pt[G]\hskip-.7pt]$ and $\Omega[\hskip-.7pt[G]\hskip-.7pt] := \mathbb{F}_p[\hskip-.7pt[G]\hskip-.7pt]$ we show that the natural map $K_1(\Lambda(G)) \to K_1(\Omega(G))$ has a splitting provided that $SK_1(\Lambda(G))$ vanishes. The image of this splitting is described in terms of a generalised norm operator. This result generalises classical work of Coleman for the case $G=\mathbb{Z}_p$. We verify the vanishing condition for certain unipotent compact $p$-adic Lie groups.

Keywords: Completed group ring, Iwasawa algebra, algebraic K-group, Adams operator, norm operator, Coleman isomorphism

Schneider Peter, Venjakob Otmar: A splitting for K1 of completed group rings. Comment. Math. Helv. 88 (2013), 613-642. doi: 10.4171/CMH/298