Tachikawa's second conjecture for self-injective algebras
Let be a finite-dimensional self-injective algebra over a field and a finite-dimensional -module that is self-orthogonal, meaning for every . Tachikawa's second conjecture, from his 1973 monograph on quasi-Frobenius rings, asserts that such an must be projective. It is the self-injective case of the Auslander-Reiten conjecture and is tied to the Nakayama conjecture through dominant dimension and endomorphism algebras. Partial results covered radical cube zero, quantum complete intersections and conditional stratification criteria (Chen-Fang-Xi). Is every self-orthogonal finite-dimensional module over a finite-dimensional self-injective algebra projective?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Representation theory of algebras; homological conjectures
- Posed by
- Hiroyuki Tachikawa, Quasi-Frobenius Rings and Generalizations: QF-3 and QF-1 Rings (Lecture Notes in Mathematics 351, 1973)
- Year posed
- 1973
- Years open
- 53y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims Theorem 1.1: over there are a finite-dimensional symmetric algebra and a nonprojective finite-dimensional module with for all , so Tachikawa's second conjecture fails, and is also a symmetric counterexample to the Auslander-Reiten conjecture. The conclusions persist under every field extension. It does NOT address Tachikawa's first conjecture separately, nor give characteristic-zero examples.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The Tachikawa manuscript reuses the companion's starting algebra and adds a transfer principle to a symmetric algebra.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture: a finite-dimensional symmetric (hence self-injective) algebra over with a nonprojective finite-dimensional self-orthogonal module. formalization.yaml lists ComparatorChallenges/Tachikawa.json, declaration OAI.Tachikawa.main_theorem, file OAI/RingTheory/Tachikawa/Counterexample.lean. The statement was read here: over the fraction field of there are a finite algebra with a linear isomorphism to its dual satisfying the symmetric-form identities, and a finite module that is not projective with all positive self-Ext groups trivial. That is the headline claim. lean/docs/199.md says the field-extension, endomorphism-algebra and homological consequences are not formalised. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.
Sources
- PaperCompanion: An explicit counterexample to the Auslander-Reiten conjecture
- Lean proofLean: Tachikawa counterexample (formalization.yaml file)Lean comparator statement: Tachikawa counterexample
- CodeOpenAI math release: A counterexample to Tachikawa's second conjecture
- Problem recordTachikawa (1973), Quasi-Frobenius Rings and Generalizations