A computation of $TH{H}_{\ast }(ku)$ using a gathered spectral sequence

Abstract

In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand $\ell$ of $p$-local connective complex topological K-theory ($ku$) to $ku$ itself. We leverage the relation $u^{p-1}=v_1$, where $u$ is a generator of $k{u}_{\ast }$ and $v_1$ is a generator of ${\ell }_{\ast }$, and we consider the cofiber of the multiplication by $v_1$ in $ku$, denoted $ku/v_1$. We use the morphism between the Bockstein spectral sequences of the multiplication by $v_1$ computing $TH{H}_{\ast }(\ell )$ and $TH{H}_{\ast }(ku)$; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequences for the multiplications by $v_1$ and $u$, from which we derive a complete computation of $TH{H}_{\ast }(ku)$. Our method is not only applicable to this specific problem but may also prove useful in other computations.

41 pages; revised the argument for the torsion-free extensions at the beginning of section 5.3; improved the presentation