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 $\mathrm{BP}⟨n⟩$. 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 $\mathrm{BP}⟨n⟩$. 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 $\mathrm{BP}⟨2⟩$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, Höning, 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 $\mathrm{BP}⟨3⟩$ at all primes $p\ge 7$, the first explicit computation of algebraic K-theory of an ${\mathbb{E}}_1$-ring of height $3$.

30 pages, 2 figure, comments welcome! V3: Improved exposition and strengthened results