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 $\mathrm{K}(\mathrm{BP}⟨n⟩{)}_{(p)}\to {\mathrm{L}}_{n+1}^f\mathrm{K}(\mathrm{BP}⟨n⟩{)}_{(p)}$ has bounded above fiber.

arXiv commentto appear in Annals of Mathematics; arXiv version has minor differences from published version