Redshift and multiplication for truncated Brown-Peterson spectra

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.