Redshift and multiplication for truncated Brown--Peterson spectra

Abstract

We equip $\mathrm{BP}(n)$ 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 $\mathrm{K}(\mathrm{BP}\langle n\rangle )_{(p)} \rightarrow \mathrm{L}_{n+1}^f \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.