Family 313TopologyNo LeanNot yet digested

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 ℤ

_{p}

-module, for every prime, positive height and integer degree. The same conclusion holds after

K(n)

-localizing any finite p-local spectrum.

Manuscripts in this family · 1

Sep 24, 2026·Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf

We prove that the homotopy groups of the

K(n)

-local sphere are finitely generated ℤ

_{p}

-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.

Full abstract