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Integral topological Hochschild homology of connective complex K-theory

David Jongwon Lee

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.

Integral topological Hochschild homology of connective complex K-theory

Abstract

We compute the homotopy groups of as a -module using the descent spectral sequence for the map , which is the motivic spectral sequence for 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 -page of the motivic spectral sequence computing , and we show that it degenerates at the -page.
Paper Structure (18 sections, 21 theorems, 136 equations, 1 figure)

This paper contains 18 sections, 21 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and previous works
  3. Main Results
  4. Outline
  5. Acknowledgements
  6. Notations and Conventions
  7. Preliminaries
  8. Topological Hochschild Homology
  9. Suspension Elements
  10. Descent Spectral Sequence
  11. Descent Spectral Sequence
  12. Complex K-theory spectrum
  13. Bockstein Spectral Sequences
  14. v0-Bockstein
  15. beta-Bockstein
  16. ...and 3 more sections

Key Result

Theorem 1.2.1

There is an isomorphism as $\mathbb Z[\beta]$-modules.

Figures (1)

  • Figure 1: $T(3)$ in degrees $38$ to $104$.

Theorems & Definitions (56)

  • Remark 1.1.1
  • Remark 1.1.2
  • Remark 1.1.3
  • Theorem 1.2.1
  • Theorem 1.2.2
  • Lemma 2.2.1
  • proof
  • Proposition 2.2.2
  • proof
  • Example 2.2.3
  • ...and 46 more