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A computation of $THH_*(ku)$ using a gathered spectral sequence

Maxime Chaminadour

TL;DR

The article addresses the problem of computing $\mathrm{THH}_*(ku)$ at a fixed prime by extending the known $\mathrm{THH}_*(\ell)$ results. It develops a gathered spectral sequence built from relations between multiplications by $v_1$ and $u$, mediated by the cofiber $ku/v_1$, and uses towers and octahedral-techniques to transfer differential information. The work delivers explicit computations for $\mathrm{THH}_*(ku;ku/v_1)$, the $u$-Bockstein spectral sequence, and a complete presentation of $\mathrm{THH}_*(ku)$ as a $ku_*$-module, including torsion structure and extensions, while highlighting that $\mathrm{THH}_*(ku)$ is not étale over $\mathrm{THH}_*(\ell)$. The methodology, especially the gathered spectral sequence, provides a general tool for relating spectral sequences across different multiplicative contexts and could be applied to other THH computations involving power relations of multiplicative elements.

Abstract

In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand $\ell$ of $p$-localized connective complex topological K-theory ($ku$) to THH of $ku$ itself. We leverage the relation $u^{p-1} = v_1$, where $u$ is a generator of $ku_*$ and $v_1$ is a generator of $\ell_*$, 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 sequence of the multiplication by $v_1$ computing $THH_*(\ell)$ and $THH_*(ku)$; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequence for the multiplications by $v_1$ and $u$, yielding a computation of $THH_*(ku)$. Our method is not only applicable to this specific problem but also potentially useful in other computations.

A computation of $THH_*(ku)$ using a gathered spectral sequence

TL;DR

The article addresses the problem of computing at a fixed prime by extending the known results. It develops a gathered spectral sequence built from relations between multiplications by and , mediated by the cofiber , and uses towers and octahedral-techniques to transfer differential information. The work delivers explicit computations for , the -Bockstein spectral sequence, and a complete presentation of as a -module, including torsion structure and extensions, while highlighting that is not étale over . The methodology, especially the gathered spectral sequence, provides a general tool for relating spectral sequences across different multiplicative contexts and could be applied to other THH computations involving power relations of multiplicative elements.

Abstract

In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand of -localized connective complex topological K-theory () to THH of itself. We leverage the relation , where is a generator of and is a generator of , and we consider the cofiber of the multiplication by in , denoted . We use the morphism between the Bockstein spectral sequence of the multiplication by computing and ; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequence for the multiplications by and , yielding a computation of . Our method is not only applicable to this specific problem but also potentially useful in other computations.

Paper Structure

This paper contains 21 sections, 37 theorems, 128 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Aims and presentation
  3. Notations and conventions
  4. Aknowlegdements
  5. Topological Hochschild homology
  6. Simplicial spectra and their realization
  7. Simplicial definition of THH and consequences
  8. Spectral sequences computing THH
  9. Smashing localizations and THH
  10. Review of the results on $\ell$ and $ku$
  11. The periodic case
  12. Topological Hochschild homology of $ku$ with coefficients in $H\mathbb{Z}_{(p)}$
  13. Topological Hochschild homology of $\ell$
  14. Spectral sequences from towers of spectra
  15. The octahedral axiom and consequences
  16. ...and 6 more sections

Key Result

Proposition 2.4

Figures (9)

  • Figure 1: Example of the spectral sequence $(\mathcal{B})$.
  • Figure 2: The spectral sequence $(\mathcal{T}_0^3)$ corresponding to the $(\mathcal{B})$ of \ref{['fig:ss-B']}.
  • Figure 3: The spectral sequence $(\mathcal{T}_3^7)$ corresponding to the $(\mathcal{B})$ of \ref{['fig:ss-B']}.
  • Figure 4: The $E^\infty$ page of $(\mathcal{T}_0^3)$, isomorphic to $(Y_0^3)_*$. The lines fix the degree.
  • Figure 5: The spectral sequence $({}^\phi \mathcal{B})$ corresponding to the $(\mathcal{B})$ of \ref{['fig:ss-B']}.
  • ...and 4 more figures

Theorems & Definitions (65)

  • Definition 2.2
  • Proposition 2.4: X.1.3 of elmendorf1997rings
  • Definition 2.8: IV.7.2 of elmendorf1997rings
  • Proposition 2.10: IV.7.5 of elmendorf1997rings
  • Proposition 2.12: X.1.2 and XII.1.2 of elmendorf1997rings
  • Definition 2.16: IX.2.1 of elmendorf1997rings
  • Proposition 2.18: IX.2.2 of elmendorf1997rings
  • Proposition 2.19
  • Proposition 2.23: IX.2.4 and IX.2.5 of elmendorf1997rings
  • proof
  • ...and 55 more