The Hovey-Strickland conjecture: the thick tensor ideals of dualizable K(n)-local spectra
Fix a prime and a height , and let be the homotopy category of dualizable -local spectra, with tensor product . A thick tensor ideal is a full subcategory closed under retracts, cofiber sequences, suspensions and tensoring with every object. For the localization of a finite type- spectrum generates a thick tensor ideal , giving a chain . By analogy with the Hopkins-Smith thick subcategory theorem for finite spectra, Hovey and Strickland (1999) conjectured that this chain is everything. Barthel, Heard and Naumann proved it at height two and showed it follows from Chai's invariant-ideal conjecture. For every prime and every height, are the thick tensor ideals of exactly the ideals , so that the Balmer spectrum is a chain of points?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Chromatic homotopy theory; tensor-triangular geometry
- Posed by
- Mark Hovey and Neil Strickland (Morava K-theories and localisation, Mem. AMS 1999, Section 12); restated by Barthel, Heard and Naumann (2022)
- Year posed
- 1999
- Years open
- 27y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.2: for every prime and the thick tensor ideals of the dualizable -local category are exactly , so there are of them and each does not depend on the chosen type- spectrum; a corollary identifies the Balmer spectrum with the proper members of the chain, with its specialization order. It is deduced from Theorem 1.1 (Chai's conjecture, a separate entry) by the known Barthel-Heard-Naumann implication. It concerns dualizable objects only and does not classify localizing or other subcategories of the whole -local category. Height two was already known.
What the AI did
The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. One manuscript proves Chai's invariant-ideal conjecture as its main step and deduces the Hovey-Strickland conjecture from it through the implication already established by Barthel, Heard and Naumann; the same manuscript underlies the separate catalog entry for Chai's conjecture.
Verification
No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 (stabilizer-invariant ideals) and Theorem 1.2 (Hovey-Strickland conjecture) were read against the conjecture as Hovey-Strickland and Barthel-Heard-Naumann state it. The release has no Lean formalization for this family (no lean/docs page at the pinned commit). The spectral classification is not proved from scratch: it uses the Barthel-Heard-Naumann implication from invariant ideals to thick tensor ideals (their Theorems 4.9 and 4.15), resting on Mathew's descent and Balmer's nilpotence criterion. The new input is the classification of invariant primes in characteristic p via perfectoid and Fargues-Fontaine curve methods; characteristic-zero primes use the Gross-Hopkins period map. The proof was not refereed here.