Syntomic cohomology of truncated Brown--Peterson spectra

We compute the $\mathrm{MU}$-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n})$, of all $\mathbb{E}1$ $\mathrm{MU}$-algebra forms of the truncated Brown–Peterson spectrum $\mathrm{BP}\langle n\rangle$. As qualitative consequences, we resolve the Lichtenbaum–Quillen, telescope, and redshift questions for the algebraic K-theories of all $\mathbb{E}{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP} \langle n\rangle$. This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 2\rangle$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, Höning, and Rognes.

💡 Research Summary

The paper studies MU‑based syntomic cohomology for all E₁‑MU‑algebra forms of the truncated Brown–Peterson spectra BP⟨n⟩, and uses these calculations to settle three major conjectural phenomena in algebraic K‑theory: the Lichtenbaum–Quillen conjecture, the telescope conjecture, and the red‑shift conjecture.

The authors begin by fixing an arbitrary E₁‑MU‑algebra R whose homotopy groups are isomorphic to ℤ₍p₎