 
Summary: Invent math (2010) 180: 611668
DOI 10.1007/s002220100239x
On the algebraic Ktheory of the complex Ktheory
spectrum
Christian Ausoni
Received: 13 February 2009 / Accepted: 4 February 2010 / Published online: 17 March 2010
© SpringerVerlag 2010
Abstract Let p 5 be a prime, let ku be the connective complex Ktheory
spectrum, and let K(ku) be the algebraic Ktheory spectrum of ku. In this pa
per we study the pprimary 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
element".
Mathematics Subject Classification (2000) 19D55 · 55N15
1 Introduction
The algebraic Ktheory 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 Ktheory, and is an
example of v1periodicity in the sense of stable homotopy theory. For exam
