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Chromatic Higher Semiadditivity by Height Induction

Shay Ben-Moshe

TL;DR

This paper addresses proving the $ ext{∞}$-semiadditivity of the category of $K(n)$-local spectra, extending the foundational Hopkins--Lurie result. It develops a height-induction approach using algebraic $K$-theory and the redshift conjecture to propagate semiadditivity across chromatic heights, avoiding the Ravenel--Wilson computation of Morava $K$-theory for Eilenberg--MacLane spaces. The work provides a new inductive proof and strengthens the link between semiadditivity and chromatic phenomena, illustrating the versatility of height-based methods in chromatic homotopy theory. The approach has potential implications for broader computations and conceptual understanding within chromatic homotopy theory.

Abstract

We give a new proof of the $\infty$-semiadditivity of $K(n)$-local spectra. The proof proceeds by induction on the height via algebraic K-theory, utilizing recent advances in chromatic homotopy theory and the redshift conjecture, instead of using the Ravenel-Wilson computation of the Morava K-theory of Eilenberg-MacLane spaces.

Chromatic Higher Semiadditivity by Height Induction

TL;DR

This paper addresses proving the -semiadditivity of the category of -local spectra, extending the foundational Hopkins--Lurie result. It develops a height-induction approach using algebraic -theory and the redshift conjecture to propagate semiadditivity across chromatic heights, avoiding the Ravenel--Wilson computation of Morava -theory for Eilenberg--MacLane spaces. The work provides a new inductive proof and strengthens the link between semiadditivity and chromatic phenomena, illustrating the versatility of height-based methods in chromatic homotopy theory. The approach has potential implications for broader computations and conceptual understanding within chromatic homotopy theory.

Abstract

We give a new proof of the -semiadditivity of -local spectra. The proof proceeds by induction on the height via algebraic K-theory, utilizing recent advances in chromatic homotopy theory and the redshift conjecture, instead of using the Ravenel-Wilson computation of the Morava K-theory of Eilenberg-MacLane spaces.
Paper Structure (2 sections, 1 equation, 1 figure)

This paper contains 2 sections, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Background

Figures (1)

  • Figure : Sazai Hall at the Temple of the Five Hundred Arhats, by Katsushika Hokusai.