Descent and cyclotomic redshift for chromatically localized algebraic 𝐾-theory

Abstract

We prove that T ( n + 1 ) T(n+1) -localized algebraic K K -theory satisfies descent for π \pi -finite p p -group actions on stable ∞ \infty -categories of chromatic height up to n n , extending a result of Clausen–Mathew–Naumann–Noel for finite p p -groups. Using this, we show that it sends T ( n ) T(n) -local Galois extensions to T ( n + 1 ) T(n+1) -local Galois extensions. Furthermore, we show that it sends cyclotomic extensions of height n n to cyclotomic extensions of height n + 1 n+1 , extending a result of Bhatt–Clausen–Mathew for n = 0 n=0 . As a consequence, we deduce that K ( n + 1 ) K(n+1) -localized K K -theory satisfies hyperdescent along the cyclotomic tower of any T ( n ) T(n) -local ring. Counterexamples to such cyclotomic hyperdescent for T ( n + 1 ) T(n+1) -localized K K -theory were constructed by Burklund, Hahn, Levy and the third author, thereby disproving the telescope conjecture.