# 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.

## Authors

Jeremy Hahn

Massachusetts Institute of Technology, Cambridge, MA

Dylan Wilson

Harvard University, Cambridge, MA