Topological Hochschild homology, truncated Brown-Peterson spectra, and a topological Sen operator

Abstract

In this article, we study the topological Hochschild homology of ${\mathbf{E}}_3$-forms of truncated Brown-Peterson spectra, taken relative to certain Thom spectra $X(p^n)$ (introduced by Ravenel and used by Devinatz-Hopkins-Smith in the proof of the nilpotence theorem). We prove analogues of Bökstedt's calculations $\mathrm{THH}({\mathbf{F}}_p)\simeq {\mathbf{F}}_p[\Omega S^3]$ and $\mathrm{THH}({\mathbf{Z}}_p)\simeq {\mathbf{Z}}_p[\Omega S^3⟨3⟩]$. We also construct a topological analogue of the Sen operator of Bhatt-Lurie-Drinfeld, and study a higher chromatic extension. The behavior of these "topological Sen operators" is dictated by differentials in the Serre spectral sequence for Cohen-Moore-Neisendorfer fibrations.

99 pages, one figure. The present article is far from being a final version, so any comments and suggestions for improvement are greatly appreciated. I'll post major updates to the arXiv, but I'll upload minor edits to my website; so please see there for the most up-to-date version

Cited by in corpus (1)

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