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

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.

💡 Research Summary

The paper investigates the p‑primary homotopy type of the algebraic K‑theory spectrum K(ku) of the connective complex K‑theory spectrum ku, for odd primes p > 3. The authors adopt a (p, v₁)‑local approach, which isolates the first chromatic layer of the stable homotopy category, and compute the mod (p, v₁) homotopy groups of K(ku). Their main result is that, after discarding a finite summand, these groups form a finitely generated free module over the polynomial algebra Fₚ