A splitting for $K_1$ of completed group rings

Abstract

For $p\ne 2$ and a uniform pro-$p$ group $G$ and its Iwasawa algebras $\Lambda (G):={\mathbb{Z}}_p[[G]]$ and $\Omega [[G]]:={\mathbb{F}}_p[[G]]$ we show that the natural map $K_1(\Lambda (G))\to K_1(\Omega (G))$ has a splitting provided that $S{K}_1(\Lambda (G))$ 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.