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 ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after $K(n)$-localizing any finite p-local spectrum.
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.