We prove that T(n+1)-localized algebraic K-theory satisfies descent for
Ο-finite p-group actions on stable β-categories of chromatic
height up to n, extending a result of Clausen-Mathew-Naumann-Noel for
p-groups. Using this, we show that it sends T(n)-local Galois extensions to
T(n+1)-local Galois extensions. Furthermore, we show that it sends cyclotomic
extensions of height n to cyclotomic extensions of height n+1, extending a
result of Bhatt-Clausen-Mathew for n=0. As a consequence, we deduce that
K(n+1)-localized K-theory satisfies hyperdescent along the cyclotomic tower
of any T(n)-local ring. Counterexamples to such cyclotomic hyperdescent for
T(n+1)-localized K-theory were constructed by Burklund, Hahn, Levy and the
third author, thereby disproving the telescope conjecture.Comment: 66 pages, comments are welcom