## Geometry & Topology

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

Christian Schlichtkrull

#### Abstract

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 $πi−1R$ not annihilated by the stable Hopf map $η∈π1s(S0)$ give rise to non-trivial classes in $Ki(R)$ for $i≥3$.

#### Article information

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

Dates
Revised: 21 April 2004
Accepted: 13 March 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883412

Digital Object Identifier
doi:10.2140/gt.2004.8.645

Mathematical Reviews number (MathSciNet)
MR2057776

Zentralblatt MATH identifier
1052.19001

#### Citation

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. https://projecteuclid.org/euclid.gt/1513883412

#### References

• C Ausoni, J Rognes, Algebraic K-theory of topological K-theory, Acta. Math. 188 (2002) 1–39
• N A Baas, B I Dundas, J Rognes, Two-vector bundles and forms of elliptic cohomology, to appear in: “Topology, Geometry and Quantum Field Theory (Proceedings of the Segal Birthday Conference)”, Cambridge University Press
• M Barratt, P Eccles, $\Gamma^+$-structures-I:A free group functor for stable homotopy theory, Topology 13 (1974) 25–45
• A K Bousfield, E M Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, from: “Geometric applications of homotopy theory (Proc. Conf. Evanstone, Ill, 1977) II”, (M G Barratt, M E Mahowald editors) Springer Lecture Notes in Math. vol. 658, Springer, Berlin (1978) 80-130
• A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Springer Lecture Notes in Math. Vol. 304, Springer, Berlin (1972)
• M Bökstedt, Topological Hochschild homology, preprint, Bielefeld (1985)
• M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993) 465–540
• M Bökstedt, F Waldhausen, The map $BSG\to A(*)\to Q(S^0)$, from: “Algebraic topology and algebraic K-Theory, (Princeton, N. J, 1983)” Ann. of Math. Stud. 113, Princeton University Press (1987) 418–431
• M Brun, Topological Hochschild homology of $\mathbb Z/p^n$, J. Pure Appl. Algebra 148 (2000) 29–76
• B Dundas, The cyclotomic trace for symmetric monoidal categories, from: “Geometry and topology: Aarhus (1996)” Contemp. Math. 258 (2000) 121–143
• B Dundas, R McCarthy, Topological Hochschild homology of ring functors and exact categories, J. Pure Appl. Algebra 109 (1996) 231–294
• M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
• J D S Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987) 403–423
• S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag (1971)
• I Madsen, Algebraic K-theory and traces, from: “Current developments in mathematics”, Internat. Press, Cambridge, MA (1994) 191–321
• M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
• J P May, The geometry of iterated loop spaces, Springer Lecture Notes in Math. vol. 271, Springer, Berlin (1972)
• J P May, F Quinn, N Ray, with contributions by J Tornehave, $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra, Lecture Notes in Math. vol. 577, Springer, Berlin (1977)
• G Segal, Categories and cohomology theories, Topology 13 (1974) 293-312
• F Waldhausen, Algebraic K-theory of topological spaces I, from: “Algebraic and geometric topology (Proc. Symp. Pure Math. Stanford Calif. 1976) Part 1”, Proc. Sympos. Pure Math. vol. 32 (1978) 35–60
• F Waldhausen, Algebraic K-theory of spaces, a manifold approach, from: “Current trends in algebraic topology, Part 1 (London, Ont, 1961)” CMS Conf Proc. 2 (1982) 141–184
• G W Whitehead, Elements of homotopy theory, Graduate Text in Mathematics, vol. 61, Springer-Verlag (1978)