A splitting for $K_1$ of completed group rings

Abstract

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.