Maxime Chaminadour | Searcharxiv

arXiv2026

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

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 $ku_*$ 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 sequences 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 sequences for the multiplications by $v_1$ and $u$, from which we derive a complete computation of $THH_*(ku)$. Our method is not only applicable to this specific problem but may also prove useful in other computations.

math.AT↗

arXiv2026

Topological Hochschild homology of truncated Brown-Peterson spectra II

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 $n\ge 2$.

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