Chromatic Noshift

TL;DR

This paper demonstrates that the chromatic redshift philosophy fails in certain rigid symmetric monoidal stable $$-categories, providing explicit examples and analyzing the implications for algebraic $K$-theory and categorification.

Contribution

It constructs examples of rigid $T(n)$-local categories where the redshift philosophy does not hold, and proves this phenomenon for a broad class of categories, extending to rational and categorification contexts.

Findings

The redshift philosophy fails in specific rigid $T(n)$-local categories.

An equivalence $K(C) o ext{End}( extbf{1}_C)^{BS^1}$ can cause $K(C)$ to vanish $T(n+1)$-locally.

The results generalize a conjecture of Levy on categorification of rings.

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.