Generalized Thom Spectra, Conjugate Stacks, and truncated Brown-Peterson Spectra

Abstract

We propose a of definition conjugate stack for a class of suitable ${\mathbb{E}}_3$-$\mathrm{MU}$-algebras that includes the truncated Brown-Peterson spectra $\mathrm{BP}⟨n⟩$ for all heights $n$. Then we calculate part of $\mathrm{BP}⟨n{⟩}^{\mathrm{conj}}$ as a quotient stack in terms of Witt vectors. To perform this computation we study generalized Thom spectra and their relative Topological Hochschild Homology. We exhibit the Eilenberg-Mac Lane spectrum $\mathrm{H}{\mathbb{F}}_2$ as a ${\mathbb{E}}_3$-$\mathrm{MU}$-Thom spectrum, the $p$-local Eilenberg-Mac Lane spectra $\mathrm{H}\mathbb{Z}/p^n$ (except for $\mathrm{H}\mathbb{Z}/4$) and $\mathrm{H}{\mathbb{Z}}_{(p)}$ as ${\mathbb{E}}_3$-${\mathrm{MU}}_{(p)}$-Thom spectra, and the connective complex K-theory spectrum $ku$ as ${\mathbb{E}}_1$-$\mathrm{MU}$-Thom spectra. We accompany these ``go" results with ``no-go" results that show these ``go" results do not admit ${\mathbb{E}}_{\infty }$-refinements.

arXiv comment31 pages