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.
Funding Statement
The first author was supported in part by a NSF postdoctoral fellowship.
The second author was supported in part by NSF grant DMS-0504069.
Citation
Andrew J. Blumberg. Michael A. Mandell. "The localization sequence for the algebraic K-theory of topological K-theory." Acta Math. 200 (2) 155 - 179, 2008. https://doi.org/10.1007/s11511-008-0025-4
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