Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the K(n)-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after K(n)-localizing any finite p-local spectrum.
Manuscript
Finite generation for the K(n)-local sphere
September 24, 2026 · source · cite as \cite{OAI:Finite-generation-for-the-Kn-local-sphere-September-24-2026}
We prove that the homotopy groups of the K(n)-local sphere are finitely generated ℤp-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after K(n)-localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.
Result 313 · Topology. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md