
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 -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…
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