Finite generation for the \(K(n)\)-local sphere

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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.

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The paper states

Theorem 1.1

In Finite generation for the K(n)-local sphere, 24 September 2026

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For every prime p, every integer n\geq 1, and every integer t, the \Zp-module

\pi_t L_{K(n)}S_p^\wedge

is finitely generated. Equivalently, for every finite p-local spectrum F, the module \pi_t L_{K(n)}F is finitely generated over \Zp in every integer degree t.

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Finite generation for the K(n)-local sphere

24 September 2026, current edition.

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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.

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