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Invent math (2010) 180: 611668 DOI 10.1007/s00222-010-0239-x

Summary: Invent math (2010) 180: 611­668
DOI 10.1007/s00222-010-0239-x
On the algebraic K-theory of the complex K-theory
Christian Ausoni
Received: 13 February 2009 / Accepted: 4 February 2010 / Published online: 17 March 2010
© Springer-Verlag 2010
Abstract Let p 5 be a prime, let ku be the connective complex K-theory
spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. In this pa-
per we study the p-primary homotopy type of the spectrum K(ku) by comput-
ing its mod (p,v1) homotopy groups. We show that up to a finite summand,
these groups form a finitely generated free module over the polynomial al-
gebra Fp[b], where b is a class of degree 2p + 2 defined as a "higher Bott
Mathematics Subject Classification (2000) 19D55 · 55N15
1 Introduction
The algebraic K-theory of a local or global number field F, with suitable
finite coefficients, is known to satisfy a form of Bott periodicity. Bott period-
icity refers here to the periodicity of topological complex K-theory, and is an
example of v1-periodicity in the sense of stable homotopy theory. For exam-


Source: Ausoni, Christian - Institut für Mathematische Statistik, Westfälische Wilhelms Universität Münster


Collections: Mathematics