Chromatic Noshift

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.