Integral topological Hochschild homology of connective complex K-theory

TL;DR

This paper computes the homotopy groups of the topological Hochschild homology of connective complex K-theory using spectral sequences and formal group law algebra, revealing a systematic approach for such computations.

Contribution

It introduces a new method to compute THH of quotients of MU via the descent spectral sequence and formal group law algebra, with explicit calculations for ku.

Findings

Homotopy groups of THH(ku) computed as a ku_* -module.

Spectral sequence for THH(ku) degenerates at E_2.

Provides a systematic algebraic approach for THH of MU quotients.

Abstract

We compute the homotopy groups of $\mathrm{THH}(\mathrm{ku})$ as a $\mathrm{ku}_\ast$-module using the descent spectral sequence for the map $\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$, which is the motivic spectral sequence for $\mathrm{THH}(\mathrm{ku})$ in the sense of Hahn-Raksit-Wilson. We reduce the computation of homotopy groups to the algebra of the universal formal group law, providing a systematic way to compute THH of quotients of MU. We compute the $E_2$-page of the motivic spectral sequence computing $\mathrm{THH}(\mathrm{ku})$, and we show that it degenerates at the $E_2$-page.