Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The monochromatic Hahn-Wilson conjecture

David Jongwon Lee, Piotr Pstrągowski

Abstract

We prove the $K(n)$-local analogue of the Hahn-Wilson conjecture on fp-spectra, which states that the truncated Brown-Peterson spectra generate the category of fp-spectra as a thick subcategory. As a corollary, we deduce the original conjecture at height $1$. Along the way, we prove the existence of $K(n)$-local finite complexes with particularly regular rings of homotopy groups.

The monochromatic Hahn-Wilson conjecture

Abstract

We prove the -local analogue of the Hahn-Wilson conjecture on fp-spectra, which states that the truncated Brown-Peterson spectra generate the category of fp-spectra as a thick subcategory. As a corollary, we deduce the original conjecture at height . Along the way, we prove the existence of -local finite complexes with particularly regular rings of homotopy groups.

Paper Structure

This paper contains 11 sections, 26 theorems, 94 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation: The Hahn-Wilson conjecture
  3. Monochromatic fp-spectra
  4. Homotopy of K(n)-local Moore spectra
  5. Acknowledgements
  6. Algebraic preliminaries
  7. Cohomology of uniform groups
  8. Twisted group algebras
  9. Homotopy of K(n)-local finite complexes
  10. Characterization of locally fp-spectra
  11. The height one case of the Hahn-Wilson conjecture

Key Result

Theorem 1.3

The Hahn-Wilson conjecture is true when $n = 1$; that is, spectra of fp-type $1$ coincide with the thick subcategory generated by $\mathrm{BP} \langle 1 \rangle$.

Theorems & Definitions (61)

  • Conjecture 1.1: Hahn-Wilson
  • Conjecture 1.2: Mahowald-Rezk, mahowald1999brown
  • Theorem 1.3: \ref{['theorem:main_text_hahn_wilson_conjecture_at_height_one']}
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7
  • Theorem 1.8: \ref{['theorem:classification_of_local_fp_types_in_terms_of_cohomology_and_morava_e_theory']}
  • Theorem 1.9: \ref{['theorem:existence_of_a_convenient_generalized_moore_and_open_subgroup']}
  • Definition 2.2
  • ...and 51 more