Syntomic cohomology of truncated Brown--Peterson spectra

TL;DR

This paper computes syntomic cohomology and algebraic K-theory of truncated Brown--Peterson spectra, resolving major conjectures and extending previous computations for specific cases at various primes.

Contribution

It provides explicit computations of syntomic cohomology and algebraic K-theory for all $ ext{MU}$-algebra forms of $ ext{BP}raket n$, addressing longstanding questions in the field.

Findings

Resolved Lichtenbaum--Quillen, telescope, and redshift conjectures.

Explicit algebraic K-theory calculations for $ ext{BP}raket 2$ at primes $p extgreater 5$.

Extended previous work to all primes $p extgreater 5$.

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.