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A note on higher topological Hochschild homology

Rixin Fang

Abstract

Chromatic redshift phenomena suggest that algebraic K-theory increases the height of a commutative ring spectrum by one. In many cases, the chromatic redshift is already detected by negative topological cyclic homology. This paper explores higher chromatic redshift via homotopy fixed points spectrum of higher topological Hochschild homology. Specifically, starting from a commutative ring spectrum that detects $v_n$-elements, the homotopy fixed points spectrum of higher topological Hochschild homology of it detects $v_{n+k}$-elements, with $k$ greater than one.

A note on higher topological Hochschild homology

Abstract

Chromatic redshift phenomena suggest that algebraic K-theory increases the height of a commutative ring spectrum by one. In many cases, the chromatic redshift is already detected by negative topological cyclic homology. This paper explores higher chromatic redshift via homotopy fixed points spectrum of higher topological Hochschild homology. Specifically, starting from a commutative ring spectrum that detects -elements, the homotopy fixed points spectrum of higher topological Hochschild homology of it detects -elements, with greater than one.
Paper Structure (7 sections, 18 theorems, 71 equations)

This paper contains 7 sections, 18 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline of the paper
  4. Acknowledgment
  5. Loday construction and higher topological Hochschild homology
  6. Homotopy fixed points of higher topological Hochschild homology
  7. Further direction

Key Result

Theorem 1.2

Let $\mathop{\mathrm{BP}}\nolimits\langle n\rangle$ be the $\mathbb{E}_3$-algebra forms constructed in HW22. Then

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.2: HW22
  • Remark 1.3
  • Theorem 1.4: Ausoniku
  • Theorem 1.5: Veen18
  • Theorem 1.7: \ref{['thm:higherredshift']}
  • Definition 2.2
  • Theorem 2.3: MSV97
  • Theorem 2.4
  • proof
  • ...and 30 more