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Syntomic cohomology of truncated Brown--Peterson spectra

Gabriel Angelini-Knoll

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, Höning, and Rognes.

Syntomic cohomology of truncated Brown--Peterson spectra

Abstract

We compute the -based syntomic cohomologies, mod , of all -algebra forms of the truncated Brown--Peterson spectrum . As qualitative consequences, we resolve the Lichtenbaum--Quillen, telescope, and redshift questions for the algebraic K-theories of all -algebra forms of . This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary -algebra forms of at all primes extending previous work of the author, Ausoni, Culver, Höning, and Rognes.
Paper Structure (15 sections, 22 theorems, 97 equations, 1 figure)

This paper contains 15 sections, 22 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Hochschild homology
  4. Forms of BPn
  5. The motivic filtration
  6. Hochschild homology with Fp-coefficients
  7. The motivic filtration on Hochschild homology
  8. Hodge Tate Cohomology
  9. Prismatic cohomology
  10. Syntomic cohomology
  11. Algebraic K-theory
  12. Algebraic K-theory at low heights
  13. Lichtenbaum--Quillen
  14. Telescope
  15. Redshift

Key Result

Theorem A

Let $p$ be a prime number and $n\ge -1$ be an integer. Let $\mathop{\mathrm{BP}}\nolimits\langle n\rangle$ be an arbitrary $\mathop{\mathrm{\mathbb{E}}}\nolimits_1$$\mathop{\mathrm{MU}}\nolimits$-algebra form of the $n$-th truncated Brown--Peterson spectra at the prime $p$, see form def. Then

Figures (1)

  • Figure 1: The mod $(2,v_1,v_2,v_3)$-syntomic cohomology of $\mathop{\mathrm{BP}}\nolimits\langle 2\rangle$

Theorems & Definitions (52)

  • Theorem A: \ref{['cor:LQC-K-theory']}, \ref{['cor:telescope']} and \ref{['cor:redshift']}
  • Theorem B: \ref{['thm:syntomic-BPn']} and \ref{['mod-(p ,...,v_n) syntomic']}
  • Theorem C: \ref{['TCBP2']} and \ref{['KBP2']}
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • ...and 42 more