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A General Blue-Shift Phenomenon

Yangyang Ruan

TL;DR

This work proposes a general blue-shift phenomenon (GBSP) in chromatic homotopy theory by examining the generalized Tate construction $ extscr{T}_{G,N}(E)$ and the roots of $[p^j]_E(x)$ in $igpi_*( extscr{T}_{G,N}(E))$. By translating series roots into Weierstrass polynomials and inverting Euler classes, the authors provide a concrete, algebraic handle on chromatic height-shifting, proving that for abelian $p$-groups the generalized Tate construction lowers the Bousfield class to $igbraket{E(n- ext{rank}_p(C))}$ and yields $v_{n- ext{rank}_p(C)}$-periodicity. They develop a robust framework connecting roots and coefficients of polynomials over commutative rings, derive vanishing results, and extend and unify prior blue-shift results (Balmer–Sanders, Ando–Morava–Sadofsky, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton). The approach leverages commutative algebra and HKR-style root data to illuminate chromatic phenomena and provides a pathway toward non-abelian extensions, with potential applications to Balmer spectrum calculations. Overall, the paper deepens the algebraic understanding of height-shifting in Tate-type constructions and strengthens the connection between chromatic homotopy theory and classical commutative algebra.

Abstract

In chromatic homotopy theory, there is a well-known conjecture called blue-shift phenomenon (BSP). In this paper, we propose a general blue-shift phenomenon (GBSP) which unifies BSP and a new variant of BSP introduced by Balmer--Sanders under one framework. To explain GBSP, we use the roots of $p^j$-series of the formal group law of a complex-oriented spectrum $E$ in the homotopy group of the generalized Tate spectrum of $E$. We also incorporate the relationship between roots and coefficients of a polynomial in any commutative ring. With this fresh perspective, we successfully achieve our goal of explaining GBSP for certain abelian cases, which provides the first example of Tate blue-shift with height-shifting at arbitrary positive integer in this setting. Additionally, we establish that the generalized Tate construction lowers Bousfield class, along with numerous Tate vanishing results. These findings strengthen and extend previous theorems of Balmer--Sanders and Ando--Morava--Sadofsky, and reproduce a result of Barthel--Hausmann--Naumann--Nikolaus--Noel--Stapleton. Furthermore, our approach simplifies the original proof of a result of Bonventre--Guillou--Stapleton, indicating that its applications are not limited to GBSP. Our work pioneers the use of commutative algebra to explain the chromatic height-shifting behavior in the blue-shift phenomenon.

A General Blue-Shift Phenomenon

TL;DR

This work proposes a general blue-shift phenomenon (GBSP) in chromatic homotopy theory by examining the generalized Tate construction and the roots of in . By translating series roots into Weierstrass polynomials and inverting Euler classes, the authors provide a concrete, algebraic handle on chromatic height-shifting, proving that for abelian -groups the generalized Tate construction lowers the Bousfield class to and yields -periodicity. They develop a robust framework connecting roots and coefficients of polynomials over commutative rings, derive vanishing results, and extend and unify prior blue-shift results (Balmer–Sanders, Ando–Morava–Sadofsky, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton). The approach leverages commutative algebra and HKR-style root data to illuminate chromatic phenomena and provides a pathway toward non-abelian extensions, with potential applications to Balmer spectrum calculations. Overall, the paper deepens the algebraic understanding of height-shifting in Tate-type constructions and strengthens the connection between chromatic homotopy theory and classical commutative algebra.

Abstract

In chromatic homotopy theory, there is a well-known conjecture called blue-shift phenomenon (BSP). In this paper, we propose a general blue-shift phenomenon (GBSP) which unifies BSP and a new variant of BSP introduced by Balmer--Sanders under one framework. To explain GBSP, we use the roots of -series of the formal group law of a complex-oriented spectrum in the homotopy group of the generalized Tate spectrum of . We also incorporate the relationship between roots and coefficients of a polynomial in any commutative ring. With this fresh perspective, we successfully achieve our goal of explaining GBSP for certain abelian cases, which provides the first example of Tate blue-shift with height-shifting at arbitrary positive integer in this setting. Additionally, we establish that the generalized Tate construction lowers Bousfield class, along with numerous Tate vanishing results. These findings strengthen and extend previous theorems of Balmer--Sanders and Ando--Morava--Sadofsky, and reproduce a result of Barthel--Hausmann--Naumann--Nikolaus--Noel--Stapleton. Furthermore, our approach simplifies the original proof of a result of Bonventre--Guillou--Stapleton, indicating that its applications are not limited to GBSP. Our work pioneers the use of commutative algebra to explain the chromatic height-shifting behavior in the blue-shift phenomenon.
Paper Structure (24 sections, 76 theorems, 190 equations)

This paper contains 24 sections, 76 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background of the blue-shift phenomenon and New tools to settle Conjecture \ref{['Gbsp']}
  4. Proof strategy of Theorem \ref{['Gtclbc']}
  5. Furture work: some ideas to settle the non-abelian cases of Conjecture \ref{['Gbsp']}
  6. Towards computing the Zariski topology of $\mathop{\mathrm{Spc}}\nolimits(\mathop{\mathrm{SH}}\nolimits(G)^c)$
  7. Review of the computation of the Zariski topology of $\mathop{\mathrm{Spc}}\nolimits(\mathop{\mathrm{SH}}\nolimits(G)^c)$
  8. Comparison between our new approach and the previous approach
  9. The homotopy groups $\pi_*(\mathscr{T}_{A,C}(E))$ and their maps
  10. The $E^*$-cohomology of the classifying space of a finite abelian $p$-group
  11. Euler classes and formal groups
  12. Maps between $E^*$-cohomology of classifying spaces
  13. The homotopy groups $\pi_*(\mathscr{T}_{A,C}(E))$
  14. Generalized relations between roots and coefficients of a polynomial
  15. Basic concepts
  16. ...and 9 more sections

Key Result

Theorem 1.4

Let $E$ be a $p$-complete, complex oriented spectrum with an associated formal group of height $n$. Let $A$ be a finite abelian $p$-group and $C$ be its direct summand. If $E$ is $\textit{Landweber}$$\textit{exact}$See La76 or Proposition LEdef for details., then $\mathscr{T}_{A,C}(E)$ is Landweber

Theorems & Definitions (143)

  • Conjecture 1.1: $\mathbf{Classical}$ $\mathbf{blue}$-$\mathbf{shift}$ $\mathbf{phenomenon}$
  • Conjecture 1.2: $\mathbf{General}$ $\mathbf{blue}$-$\mathbf{shift}$ $\mathbf{phenomenon}$
  • Remark 1.3
  • Theorem 1.4: $\mathbf{Generalized}$ $\mathbf{Tate}$ $\mathbf{construction}$ $\mathbf{lowers}$ $\mathbf{Bousfield}$ $\mathbf{class}$
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.11
  • ...and 133 more