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Syntomic cohomology of Morava K-theory

Gabriel Angelini-Knoll, Jeremy Hahn, Dylan Wilson

TL;DR

We develop MU-based syntomic cohomology techniques to study connective Morava K-theory $k(n)$ and its algebraic K-theory via $K(n)$, establishing redshift results and telescope-type conjectures for the associated Morava theories. The core method combines THH/TC with a motivic filtration relative to $MU$ and a Nygaard-completed prismatic/syntomic framework, yielding a complete, finite description of mod $(p,v_1, abla, abla)$ syntomic data for $k(n)$ and a clear, three-line concentration phenomenon in the motivic TC spectral sequence. The main technical breakthrough is the explicit computation of mod $(p,v_1, abla)$ syntomic cohomology of $k(n)$, including the construction of classes $\lambda_k$, relations among $\varepsilon_i$ and $\overline{\varepsilon}_i$, and a controlled pattern of $t$-differentials that collapse the motivic spectral sequences. Collectively, these results illuminate the structure of algebraic K-theory for Morava K-theories, provide finiteness statements for syntomic data, and support the identification of $\mathrm{K}(K(n))$ in terms of its chromatic height, with implications for redshift, Lichtenbaum–Quillen properties, and telescope conjectures.

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 $π_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$.

Syntomic cohomology of Morava K-theory

TL;DR

We develop MU-based syntomic cohomology techniques to study connective Morava K-theory and its algebraic K-theory via , establishing redshift results and telescope-type conjectures for the associated Morava theories. The core method combines THH/TC with a motivic filtration relative to and a Nygaard-completed prismatic/syntomic framework, yielding a complete, finite description of mod syntomic data for and a clear, three-line concentration phenomenon in the motivic TC spectral sequence. The main technical breakthrough is the explicit computation of mod syntomic cohomology of , including the construction of classes , relations among and , and a controlled pattern of -differentials that collapse the motivic spectral sequences. Collectively, these results illuminate the structure of algebraic K-theory for Morava K-theories, provide finiteness statements for syntomic data, and support the identification of in terms of its chromatic height, with implications for redshift, Lichtenbaum–Quillen properties, and telescope conjectures.

Abstract

We compute the MU-based syntomic cohomologies, mod , of all -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 --algebra forms of -periodic Morava K-theory. Notably, the motivic spectral sequence computing is concentrated on at most three lines, independently of .

Paper Structure

This paper contains 40 sections, 74 theorems, 444 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Three qualitative questions
  3. Detailed results
  4. Acknowledgements
  5. Hochschild homology and the motivic filtration
  6. Structures on Hochschild homology
  7. The motivic filtration
  8. The sigma operator
  9. The cap product
  10. The double suspension map
  11. Hochschild homology with $\mathop{\mathrm{\mathbb{F}}}\nolimits_p$ coefficients
  12. Hochschild homology with $\mathop{\mathrm{\mathbb{F}}}\nolimits_p$-coefficients
  13. The motivic filtration with $\mathop{\mathrm{\mathbb{F}}}\nolimits_p$ coefficients
  14. The mod $p$ motivic filtrations on $\mathop{\mathrm{THH}}\nolimits(\mathop{\mathrm{MU}}\nolimits)$ and $\mathop{\mathrm{THH}}\nolimits(\mathop{\mathrm{BP}}\nolimits)$
  15. Mod $(p,v_1,\cdots,v_n)$ Hochschild homology of $\mathbb{F}_p$
  16. ...and 25 more sections

Key Result

Theorem A

The spectrum $\mathop{\mathrm{K}}\nolimits(K(n))$ has height exactly $n+1$.

Figures (5)

  • Figure 1.2.1: The mod $(2,v_1,v_2,v_3)$-syntomic cohomology of $k(2)$.
  • Figure 3.4.1: The $\mathrm{E}_\infty$-page of the periodic t-Bockstein spectral sequence computing the mod $(2,v_1,v_2,v_3)$-prismatic cohomology of $k(2)$. Each named class is a generator of a copy of $\mathbb{F}_2$.
  • Figure 3.5.1: The $\mathrm{E}_\infty$-page of the $t$-Bockstein spectral sequence converging to $\pi_{*}\mathrm{gr}_{\textup{mot}}^{*}\mathop{\mathrm{TC}}\nolimits^{-}(k(2))/(2,v_{1},v_{2},v_{3})$. Each named class represents a copy of $\mathbb{F}_2$. We draw $M_{\emptyset}$ in blue, $M_{1}$ in green, $M_{2}$ in orange, and $M_{\{1,2\}}$ in red.
  • Figure 3.5.2: The mod $(3,v_1,v_2)$-syntomic cohomology of $k(1)$
  • Figure 4.5.1: The mod $(2,v_1,v_2,v_3)$-syntomic cohomology of $k_{\bar{\mathop{\mathrm{\mathbb{F}}}\nolimits}_{2}}(2)$. Here boxed classes are generators of $\mathop{\mathrm{\mathbb{F}}}\nolimits_2$ and unboxed classes are generators of $\bar{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2$

Theorems & Definitions (195)

  • Definition 1.2.1
  • Theorem A: Redshift
  • Theorem B: Lichtenbaum--Quillen
  • Theorem C: Telescope
  • Remark 1.2.2
  • Theorem D: Pure fp-type
  • Theorem E: Finite syntomic cohomology
  • Remark 1.2.3
  • Remark 1.2.4
  • Remark 1.2.5
  • ...and 185 more