Geometry & Topology

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

Christian Schlichtkrull

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Let GL1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition

η R : B G L 1 ( 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 πi1R not annihilated by the stable Hopf map ηπ1s(S0) give rise to non-trivial classes in Ki(R) for i3.

Article information

Geom. Topol., Volume 8, Number 2 (2004), 645-673.

Received: 25 November 2003
Revised: 21 April 2004
Accepted: 13 March 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 19D10: Algebraic $K$-theory of spaces 55P48: Loop space machines, operads [See also 18D50]

ring spectra algebraic K-theory topological Hochschild homology


Schlichtkrull, Christian. Units of ring spectra and their traces in algebraic $K$–theory. Geom. Topol. 8 (2004), no. 2, 645--673. doi:10.2140/gt.2004.8.645.

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