JournalscmhVol. 88, No. 3pp. 613–642

A splitting for <em>K</em><sub>1</sub> of completed group rings

  • Peter Schneider

    Universität Münster, Germany
  • Otmar Venjakob

    Universität Heidelberg, Germany
A splitting for <em>K</em><sub>1</sub> of completed group rings cover
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For p2p\neq 2 and a uniform pro-pp group GG and its Iwasawa algebras Λ(G):=Zp[[G]]\Lambda (G) := \mathbb{Z}_{p}[\hskip-.7pt[G]\hskip-.7pt] and Ω[[G]]:=Fp[[G]]\Omega[\hskip-.7pt[G]\hskip-.7pt] := \mathbb{F}_p[\hskip-.7pt[G]\hskip-.7pt] we show that the natural map K1(Λ(G))K1(Ω(G))K_1(\Lambda(G)) \to K_1(\Omega(G)) has a splitting provided that SK1(Λ(G))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=ZpG=\mathbb{Z}_p. We verify the vanishing condition for certain unipotent compact pp-adic Lie groups.

Cite this article

Peter Schneider, Otmar Venjakob, A splitting for <em>K</em><sub>1</sub> of completed group rings. Comment. Math. Helv. 88 (2013), no. 3, pp. 613–642

DOI 10.4171/CMH/298