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

We prove some K-theoretic descent results for finite group actions on stable ∞-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 E_∞-ring spectrum with LT(n)R=0, then LT(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.