Descent and vanishing in chromatic algebraic $K$-theory via group actions

TL;DR

This paper proves descent and vanishing results in chromatic algebraic K-theory for finite group actions on stable $$-categories, confirming conjectures and illustrating the interplay between descent and vanishing through induction.

Contribution

It establishes new descent results for finite group actions, including the Galois descent conjecture, and demonstrates vanishing phenomena aligned with the redshift philosophy.

Findings

Proves the $p$-group case of the Galois descent conjecture.

Shows that vanishing of $L_{T(n)}R$ implies vanishing of $L_{T(n+1)}K(R)$.

Establishes a logical link between descent and vanishing via induction.

Abstract

We prove some $K$-theoretic descent results for finite group actions on stable $\infty$-categories, including the $p$-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if $R$ is an $\mathbb{E}_\infty$-ring spectrum with $L_{T(n)}R=0$, then $L_{T(n+1)}K(R)=0$. Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.