Finite generation for the K(n)-local sphere
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.
Cite
@misc{OAI:Finite-generation-for-the-Kn-local-sphere-September-24-2026,
author = {{OpenAI}},
title = {{Finite generation for the $K(n)$-local sphere}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf}{OAI:Finite-generation-for-the-Kn-local-sphere-September-24-2026}},
year = {2026}
}