Algebraic K-theory of topological K-theory

Let ℓp be the p-complete connective Adams summand of topological K-theory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)-homotopy of the algebraic K-theory spectrum of ℓp, denoted V (1)∗K(ℓp). In particular we find that it is a free finitely generated module over the polynomial algebra P (v2), except for a sporadic class in degree 2p − 3. Thus also in this case algebraic K-theory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is