## Acta Mathematica

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

#### Abstract

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.

#### Note

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

#### Note

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

#### Article information

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

Dates
Revised: 11 February 2007
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485891978

Digital Object Identifier
doi:10.1007/s11511-008-0025-4

Mathematical Reviews number (MathSciNet)
MR2413133

Zentralblatt MATH identifier
1149.18008

Rights

#### Citation

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. https://projecteuclid.org/euclid.acta/1485891978

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