Topological Hochschild homology of truncated Brown-Peterson spectra I

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.