Remark: Family 313. Finite generation for the K(n) -local sphere

Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the $K(n)$-local sphere is a finitely generated ${\mathbb{Z}}_p$-module, for every prime, positive height and integer degree. The same conclusion holds after $K(n)$-localizing any finite $p$-local spectrum.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.