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

Abstract

Let $G{L}_1\left(R\right)$ be the units of a commutative ring spectrum $R$. In this paper we identify the composition

$${\eta }_R:BG{L}_1\left(R\right)\to K\left(R\right)\to THH\left(R\right)\to {\Omega }^{\infty }\left(R\right),$$

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 $i\ge 3$.