# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.331992

Acta Mathematica, 200, (2), 155-179, 2006.6
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.
@inproceedings{andrew2006the,
title={The localization sequence for the algebraic$K$-theory of topological$K$-theory},
author={Andrew J. Blumberg, and Michael A. Mandell},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351707632701},
booktitle={Acta Mathematica},
volume={200},
number={2},
pages={155-179},
year={2006},
}

Andrew J. Blumberg, and Michael A. Mandell. The localization sequence for the algebraic$K$-theory of topological$K$-theory. 2006. Vol. 200. In Acta Mathematica. pp.155-179. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351707632701.