Syntomic cohomology of Morava K-theory

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 ${\pi }_{\ast }TC(k(n){)}_p$ is concentrated on at most three lines, independently of $n$.

65 pages, 5 figures, minor revisions. Comments still welcome!