#### 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.

Notes on file formats
Christian Schlichtkrull

Department of Mathematics, Oslo University

PO Box 1053, Blindern

NO-0316 Oslo, Norway

Email: krull@math.uio.no

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