Volume 8, issue 2 (2004)

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Units of ring spectra and their traces in algebraic $K$–theory

Christian Schlichtkrull

Geometry & Topology 8 (2004) 645–673
 arXiv: math.AT/0405079
Abstract

Let $G{L}_{1}\left(R\right)$ be the units of a commutative ring spectrum $R$. In this paper we identify the composition

${\eta }_{R}:\phantom{\rule{0.3em}{0ex}}BG{L}_{1}\left(R\right)\to K\left(R\right)\to THH\left(R\right)\to {\Omega }^{\infty }\left(R\right),$

where $K\left(R\right)$ is the algebraic $K$–theory and $THH\left(R\right)$ the topological Hochschild homology of $R$. As a corollary we show that classes in ${\pi }_{i-1}R$ not annihilated by the stable Hopf map $\eta \in {\pi }_{1}^{s}\left({S}^{0}\right)$ give rise to non-trivial classes in ${K}_{i}\left(R\right)$ for $i\ge 3$.

Keywords
ring spectra, algebraic K-theory, topological Hochschild homology
Mathematical Subject Classification 2000
Primary: 19D55, 55P43
Secondary: 19D10, 55P48