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Transchromatic phenomena in the equivariant slice spectral sequence

Lennart Meier, XiaoLin Danny Shi, Mingcong Zeng

Abstract

In this paper, we prove a transchromatic phenomenon for Hill--Hopkins--Ravenel and Lubin--Tate theories. This establishes a direct relationship between the equivariant slice spectral sequences of height-$h$ and height-$(h/2)$ theories. As applications of this transchromatic phenomenon, we prove periodicity and vanishing line results for these theories.

Transchromatic phenomena in the equivariant slice spectral sequence

Abstract

In this paper, we prove a transchromatic phenomenon for Hill--Hopkins--Ravenel and Lubin--Tate theories. This establishes a direct relationship between the equivariant slice spectral sequences of height- and height- theories. As applications of this transchromatic phenomenon, we prove periodicity and vanishing line results for these theories.
Paper Structure (17 sections, 30 theorems, 185 equations, 7 figures)

This paper contains 17 sections, 30 theorems, 185 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivation and main theorem
  3. Periodicities and vanishing lines
  4. The stratification tower of Hill--Hopkins--Ravenel theories
  5. Open questions and future directions
  6. Organization of the paper
  7. Stratification of the slice spectral sequence
  8. Shearing isomorphisms of spectral sequences
  9. The pullback functor
  10. The dual slice tower
  11. Shearing isomorphism
  12. Isomorphism of localized slice towers
  13. Correspondence formulas
  14. The Transchromatic Isomorphism Theorem
  15. The transchromatic tower
  16. ...and 2 more sections

Key Result

Theorem A

For $G = C_{2^{n+1}}$ and $m \geq 1$, there is a shearing isomorphism $d_{2r-1} \leftrightsquigarrow d_r$ between the following regions of spectral sequences:

Figures (7)

  • Figure 1: The Transchromatic Isomorphism Theorem.
  • Figure 2: Vanishing Lines in the slice spectral sequence.
  • Figure 3: The stratification tower of the slice spectral sequence.
  • Figure 4: The slice spectral sequence of $BP^{(\!(C_{4})\!)} \langle 1 \rangle$.
  • Figure 5: The slice spectral sequence of $BP_\mathbb{R} \langle 1 \rangle$.
  • ...and 2 more figures

Theorems & Definitions (79)

  • Conjecture 1.1: Transchromatic Phenomenon
  • Theorem A: Transchromatic Isomorphism
  • Theorem B: Periodicity, \ref{['thm:ROGPeriodicityTranschromatic']}
  • Theorem C: Vanishing Lines, \ref{['thm:VanishingLineGeneralSlope']}
  • Definition 1.2
  • Theorem D: Localized Slice Tower Isomorphism, \ref{['thm:DualTowerEquivalenceMain']}
  • Theorem E: Shearing Isomorphism, \ref{['thm:ShearingGeneralTheory']}, \ref{['thm:TranschromaticMain']}
  • Theorem F: Correspondence formulas, \ref{['thm:CorrespondenceFormula']}
  • Conjecture 1.3
  • Conjecture 1.4
  • ...and 69 more