Topological Hochschild homology of truncated Brown-Peterson spectra I

TL;DR

This paper computes the topological Hochschild homology of structured truncated Brown-Peterson spectra with coefficients, providing explicit calculations for various cases including certain primes and forms.

Contribution

It offers new explicit computations of THH for truncated Brown-Peterson spectra with coefficients, extending previous knowledge to specific structured forms and primes.

Findings

Computed THH for all n with coefficients over integers at prime p

Calculated THH for B⟨2⟩ with k(1) and k(2) coefficients

Provided explicit THH results for taf^D with various coefficients

Abstract

We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $\mathrm{THH}_*(B\langle n\rangle ;H\mathbb{Z}_{(p)})$ for all $n$ and $\mathrm{THH}_*(B\langle 2\rangle ;M)$ for $M\in \{ k(1),k(2)\}$ where $B\langle n\rangle$ is an $E_3$ form of $BP\langle n\rangle$ for certain primes $p$. For example, this gives a computation of $\mathrm{THH}(\mathrm{taf}^{D};M)$ for $M\in \{H\mathbb{Z}_{(3)},k(1),k(1))$ where $\mathrm{taf}^{D}$ is the $E_{\infty}$ form of $BP\langle 2\rangle$ constructed by Hill--Lawson.