Geometry & Topology, Vol. 8 (2004) Paper no. 16, pages 645--673.

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

Christian Schlichtkrull

Abstract. Let GL_1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition
BGL_1(R)->K(R)->THH(R)->\Omega ^{\infty }(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 \pi _{i-1}(R) not annihilated by the stable Hopf map give rise to non-trivial classes in K_i(R) for i\geq 3.

Keywords. Ring spectra, algebraic K-theory, topological Hochschild homology

AMS subject classification. Primary: 19D55, 55P43. Secondary: 19D10, 55P48.

DOI: 10.2140/gt.2004.8.645

E-print: arXiv:math.AT/0405079

Submitted to GT on 25 November 2003. (Revised 21 April 2004.) Paper accepted 13 March 2004. Paper published 22 April 2004.

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Christian Schlichtkrull
Department of Mathematics, Oslo University
PO Box 1053, Blindern
NO-0316 Oslo, Norway

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