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Chromatic Noshift

Maxime Ramzi

Abstract

The chromatic redshift philosophy, introduced by Ausoni and Rognes, suggests that algebraic $K$-theory raises chromatic height by $1$. We show that the analogue of this philosophy fails in the case of rigid symmetric monoidal stable $\infty$-categories. More precisely, we construct examples of rigid $T(n)$-local categories $C$ where a refinement $\mathrm{Dim}$ of the dimension morphism induces an equivalence $$K(C)\to \mathrm{End}(\mathbf{1}_C)^{BS^1}$$ and for which $K(C)$ therefore vanishes $T(n+1)$-locally. In fact, we prove that this equivalence always holds for $\aleph_1$-Nullstellensatzian rigid $T(n)$-local categories in the sense of Burklund, Schlank and Yuan. We study more in depth the rational version of these results to find a rigid rational additive $1$-category witnessing the failure of redshift at height $0$. Finally, we use our methods to prove and generalize a conjecture of Levy about categorification of ordinary rings.

Chromatic Noshift

Abstract

The chromatic redshift philosophy, introduced by Ausoni and Rognes, suggests that algebraic -theory raises chromatic height by . We show that the analogue of this philosophy fails in the case of rigid symmetric monoidal stable -categories. More precisely, we construct examples of rigid -local categories where a refinement of the dimension morphism induces an equivalence and for which therefore vanishes -locally. In fact, we prove that this equivalence always holds for -Nullstellensatzian rigid -local categories in the sense of Burklund, Schlank and Yuan. We study more in depth the rational version of these results to find a rigid rational additive -category witnessing the failure of redshift at height . Finally, we use our methods to prove and generalize a conjecture of Levy about categorification of ordinary rings.

Paper Structure

This paper contains 16 sections, 63 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categorical algebra
  4. Motives
  5. Dimension morphisms and the generalized 1$D$ cobordism hypothesis
  6. $K$-theory and Dimensions
  7. Nullstellensatzian rigid categories
  8. Nullstellensätze
  9. Basics about Nullstellensatzian rigid categories
  10. A $K$-theoretic Nullstellensatz
  11. Rational rigid categories and semisimplicity
  12. Dimensions in characteristic $0$
  13. Semi-simple categories
  14. Nonintegral objects
  15. Duality
  16. ...and 1 more sections

Key Result

Theorem 1

Let $C$ be a nonzero rigid $T(n)$-local symmetric monoidal $\infty$-category. There exists a nonzero rigid $T(n)$-local symmetric monoidal $\infty$-category and a symmetric monoidal functor (which can be chosen to be fully faithful) $C\to D$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Sun Setting over a Lake, J.M.W. Turner

Theorems & Definitions (183)

  • Example
  • Theorem : Noshift, \ref{['cor:noshift']}
  • Remark
  • Remark
  • Theorem : \ref{['thm:designer']}
  • Remark
  • Corollary : \ref{['cor:levytext']}
  • Theorem : $K$-theoretic Nullstellensatz, \ref{['thm:KNS', 'lm:end1NS', 'cor:NSnoshift']}
  • Example
  • Remark
  • ...and 173 more