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Finite generation for the K(n)-local sphere
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Abstract
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.
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Possibly relevant literature (found by the agent, not checked)
- P. Goerss, H.-W. Henn, M. Mahowald, C. Rezk, A resolution of the K(2)-local sphere at the prime 3 (arXiv:0706.2175; Ann. of Math. 162 (2005)). Builds a finite resolution of the K(2)-local sphere at p=3, a height-two case beyond the p≥5 finiteness results cited in the paper's history section.
- A. Beaudry, P. G. Goerss, H.-W. Henn, Chromatic splitting for the K(2)-local sphere at p=2 (arXiv:1712.08182; Geom. Topol. 26 (2022)). Computes continuous cohomology of the height-two stabilizer group with Lubin–Tate coefficients E_0 at p=2, the kind of coefficient cohomology whose finiteness (mod p) is the paper's main algebraic input.
- T. Barthel, T. M. Schlank, N. Stapleton, J. Weinstein, On Hopkins' Picard group (arXiv:2407.20958, v2 December 2025). Uses p-adic geometry (Drinfeld's symmetric space) to compute the algebraic Picard group of K(n)-local spectra at all heights and primes, bearing on the paper's remark that degreewise finite generation does not extend to arbitrary invertible K(n)-local spectra.
Digests and expositions
- Jiahao Li (UBC Okanagan), On OpenAI's Proof of Z_p-Finite Generation of K(n)-local Sphere (GitHub Pages site, 2026; source repository). An expository site, marked "(UPDATING)", whose proof sketch covers the reduction to finiteness of π_*L_{K(n)}(S/p) and to continuous stabilizer cohomology with "allowed" coefficients, deferring the proof of the coefficient theorem.
- Latent Seminar, Finite generation for the local sphere: transfers and compactness (7 October 2026). Explains the manuscript's coefficient-finiteness strategy, fixed-level transfer argument, and passage to integral homotopy via descent and compact Nakayama, explicitly disclaiming independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 313
Independent sites; their content is not reviewed here.
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}
}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 ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after $K(n)$-localizing any finite p-local spectrum.
This is the only paper in this family.