OpenAI · 313-01 · 2026-09-24 · Topology
Finite generation for the K(n)-local sphere
preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026
Family 313. Finite generation for the -local sphere.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the -local sphere is a finitely generated -module, for every prime, positive height and integer degree. The same conclusion holds after -localizing any finite -local spectrum.
We prove that the homotopy groups of the -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 -localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.