Syntomic cohomology of truncated Brown--Peterson spectra

Abstract

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, H\"oning, and Rognes.A dditionally, we present a new computation of mod $(p, v_1, v_2, v_3)$ algebraic K-theory of arbitrary $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 3\rangle$ at all primes $p\ge 7$, the first explicit computation of algebraic K-theory of an $\mathbb{E}_1$-ring of height $3$.