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Topological Hochschild homology of truncated Brown-Peterson spectra II

Gabriel Angelini-Knoll, Maxime Chaminadour

Abstract

We compute topological Hochschild homology of $\mathbb{E}_3$-MU-algebra forms of the second truncated Brown-Peterson spectrum with Adams summand coefficients at $p=2$ and conditionally at arbitrary primes. We also provide a new computational tool, a variant of the Brun spectral sequence, for computing topological Hochschild homology of truncated Brown-Peterson spectra with certain coefficients. As a consequence, we show that $\mathbb{E}_3$-MU-algebra forms of truncated Brown-Peterson spectra are not Thom spectra $\mathrm{BP}\langle n\rangle$ at the prime $p=2$ for any $n\ge 2$.

Topological Hochschild homology of truncated Brown-Peterson spectra II

Abstract

We compute topological Hochschild homology of -MU-algebra forms of the second truncated Brown-Peterson spectrum with Adams summand coefficients at and conditionally at arbitrary primes. We also provide a new computational tool, a variant of the Brun spectral sequence, for computing topological Hochschild homology of truncated Brown-Peterson spectra with certain coefficients. As a consequence, we show that -MU-algebra forms of truncated Brown-Peterson spectra are not Thom spectra at the prime for any .
Paper Structure (10 sections, 20 theorems, 108 equations)

This paper contains 10 sections, 20 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Conventions
  4. Acknowledgements
  5. Recollections
  6. Bounding Hochschild homology of BP n
  7. Hochschild homology BP2 with BP1 coefficients
  8. Computating the spectral sequence
  9. Computing the extensions
  10. Truncated Brown--Peterson spectra are not Thom spectra

Key Result

Theorem A

Suppose $p=2$ and $\mathrm{BP}\langle 2\rangle$ is a $\mathbb{E}_3$-$\mathrm{MU}$-algebra form of the second truncated Brown--Peterson spectrum or $p=3$ and $\mathrm{BP}\langle 2\rangle=\mathrm{taf}^{D}$. The topological Hochschild homology of $\mathrm{BP}\langle 2\rangle$ is a direct sum where $\mathcal{T}$ is the $\mathbb{Z}_{(p)}[v_1]$-submodule of $\mathop{\mathrm{THH}}\nolimits_*(\mathrm{BP}

Theorems & Definitions (43)

  • Theorem A
  • Theorem B
  • Theorem 2.1: AHL10
  • Theorem 2.2: ACH24*Theorem 3.8
  • Corollary 2.3
  • Theorem 2.4: AHL10
  • Proposition 2.6: ACH24*Proposition 3.7
  • Proposition 3.1
  • Lemma 3.3
  • proof
  • ...and 33 more