Redshift and multiplication for truncated Brown-Peterson spectra

TL;DR

This paper constructs an $ ext{E}_3$-algebra structure on truncated Brown-Peterson spectra, revealing their algebraic $K$-theory has chromatic height $n+1$ and analyzing related $K$-theory maps.

Contribution

It introduces an $ ext{E}_3$-algebra structure on $ ext{BP} extless n extgreater$ spectra and studies their algebraic $K$-theory properties at various primes and heights.

Findings

Algebraic $K$-theory of $ ext{BP} extless n extgreater$ has chromatic height exactly $n+1$.

The map from $K( ext{BP} extless n extgreater)$ to its chromatic localization has bounded fiber.

The construction applies for each prime $p$ and height $n$.

Abstract

We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm{K}(\mathrm{BP}\langle n \rangle)_{(p)} \to \mathrm{L}_{n+1}^{f} \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.