Units of ring spectra and their traces in algebraic K-theory

Abstract

Let GL1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition ηR: BGL1(R) → K(R) → THH(R) → Ω∞(R), where K(R) is the algebraic K-theory and THH(R) the topological Hochschild homology of R. As a corollary we show that classes in πi-1R not annihilated by the stable Hopf map η ε π18(S0) give rise to non-trivial classes in Ki(R) for i ≥ 3.