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Syntomic cohomology of Morava K-theory (2410.07048v2)
Published 9 Oct 2024 in math.KT and math.AT
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 $(2pn-2)$-periodic Morava K-theory. Notably, the motivic spectral sequence computing $\pi_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$.