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

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra (K(mathbb{Z})to K(ku)to K(KU) toSigma 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.