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

Abstract

We prove a conjecture of Rognes by establishing a localization cofiber sequence of spectra, K(Z) to K(ku) to K(KU) to Sigma K(Z), for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a devissage theorem identifying the K-theory of the Waldhausen category of Postnikov towers of modules over a connective A-infinity ring spectrum R with the Quillen K-theory of the abelian category of finitely generated pi_0(R)-modules.

Updated final version. Small change in definition of S' construction and correction to the proof of 2.9

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• Purity in chromatically localized algebraic $K$-theory
• K-theory of derivators revisited
• On topological Hochschild homology of the $K(1)$-local sphere
• Dévissage for Waldhausen K-theory