On the algebraic K-theory of the complex K-theory spectrum

TL;DR

This paper investigates the algebraic K-theory of the connective complex K-theory spectrum at primes greater than 3, revealing a structured module pattern in its mod (p,v_1) homotopy groups.

Contribution

It provides the first detailed computation of the mod (p,v_1) homotopy groups of K(ku), showing they form a finitely generated free module over a polynomial algebra.

Findings

Homotopy groups form a finitely generated free module over F_p[b]

Identification of a higher Bott element b of degree 2p+2

Structure holds up to a finite summand

Abstract

Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over a polynomial algebra F_p[b], where b is a class of degree 2p+2 defined as a higher Bott element.