Acta Mathematica

The localization sequence for the algebraic K-theory of topological K-theory

Andrew J. Blumberg and Michael A. Mandell

Full-text: Open access


We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra $K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})$ for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the K-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective $A_\infty$ ring spectrum R with the Quillen K-theory of the abelian category of finitely generated $\pi_{0}R$-modules.


The first author was supported in part by a NSF postdoctoral fellowship.


The second author was supported in part by NSF grant DMS-0504069.

Article information

Acta Math., Volume 200, Number 2 (2008), 155-179.

Received: 26 June 2006
Revised: 11 February 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D99: None of the above, but in this section
Secondary: 19L99: None of the above, but in this section 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

2008 © Institut Mittag-Leffler


Blumberg, Andrew J.; Mandell, Michael A. The localization sequence for the algebraic K -theory of topological K -theory. Acta Math. 200 (2008), no. 2, 155--179. doi:10.1007/s11511-008-0025-4.

Export citation


  • A usoni, C., Topological Hochschild homology of connective complex K-theory. Amer. J. Math., 127:6 (2005), 1261–1313.
  • A usoni, C. & R ognes, J., Algebraic K-theory of topological K-theory. Acta Math., 188 (2002), 1–39.
  • B aas, N.A., D undas, B. I. & R ognes, J., Two-vector bundles and forms of elliptic cohomology, in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., 308, pp. 18–45. Cambridge Univ. Press, Cambridge, 2004.
  • E lmendorf, A. D., K riz, I., M andell, M. A. & M ay, J.P., Rings, Modules, and Algebras in Stable Homotopy Theory. Mathematical Surveys and Monographs, 47. Amer. Math. Soc., Providence, RI, 1997.
  • M ay, J. P., Simplicial Objects in Algebraic Topology. Van Nostrand Mathematical Studies, 11. Van Nostrand, Princeton, NJ, 1967.
  • M cC lure, J. E. & S taffeldt, R. E., The chromatic convergence theorem and a tower in algebraic K-theory. Proc. Amer. Math. Soc., 118:3 (1993), 1005–1012.
  • Q uillen, D., Higher algebraic K-theory, I, in Algebraic K-theory, I: Higher K-Theories (Battelle Memorial Inst., Seattle, WA, 1972), Lecture Notes in Math., 341, pp. 85–147. Springer, Berlin–Heidelberg, 1973.
  • R ognes, J., Galois extensions of structured ring spectra. Stably dualizable groups. Mem. Amer. Math. Soc., 192:898 (2008).
  • T homason, R.W. & T robaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Vol. III, Progr. Math., 88, pp. 247–435. Birkhäuser, Boston, MA, 1990.
  • W aldhausen, F., Algebraic K-theory of spaces, a manifold approach, in Current Trends in Algebraic Topology, Part 1 (London, Ont., 1981), CMS Conf. Proc., 2, pp. 141–184. Amer. Math. Soc., Providence, RI, 1982.
  • — Algebraic K-theory of spaces, in Algebraic and Geometric Topology (New Brunswick, NJ, 1983), Lecture Notes in Math., 1126, pp. 318–419. Springer, Berlin–Heidelberg, 1985.