Redshift and multiplication for truncated Brown-Peterson spectra

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.

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