  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.
