Syntomic cohomology of Morava K-theory

TL;DR

This paper computes syntomic cohomology for Morava K-theory, leading to proofs of major conjectures in algebraic K-theory and revealing a simplified spectral sequence structure.

Contribution

It provides explicit computations of syntomic cohomology for all E₁-MU-algebra forms of connective Morava K-theory, establishing several key conjectures.

Findings

Proves the Lichtenbaum--Quillen conjecture for these algebraic K-theories.

Establishes the telescope and redshift conjectures in this context.

Shows the motivic spectral sequence is concentrated on at most three lines, regardless of n.

Abstract

We compute the MU-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n+1})$, of all $\mathbb{E}_1$-MU-algebra forms of connective Morava K-theory k(n). As qualitative consequences, we deduce the Lichtenbaum--Quillen conjecture, telescope conjecture, and redshift conjecture for the algebraic K-theories of all $\mathbb{E}_{1}$-$\mathbb{S}$-algebra forms of $(2p^n-2)$-periodic Morava K-theory. Notably, the motivic spectral sequence computing $\pi_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$.