Tachikawa’s second conjecture for symmetric algebras
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Tachikawa’s second conjecture says that a finite-dimensional module M over a finite-dimensional self-injective algebra is projective if Extⁱ(M,M)=0 for every i>0. Does this hold even for symmetric algebras?
Why it matters and what counts as progress
Why it matters: The implication would turn complete self-orthogonality into a strong structural property of the module.
What counts as progress: A proof or a finite-dimensional field-algebra witness with a nonprojective module and vanishing in every positive degree; partial Ext vanishing is insufficient.
How the problem developed
- 2024 · Structural reduction [1]
Hongxing Chen, Ming Fang, and Changchang Xi Related the conjecture for symmetric algebras to stratifying ideals and constructed potential counterexamples from possible counterexamples to Nakayama’s conjecture.
- 23 Sept 2026 · Counterexample [2]
OpenAI Claimed a finite-dimensional symmetric algebra over F₂(q,H₁,H₂) with a nonprojective finite-dimensional module whose positive self-extension groups all vanish.
AI contribution
A counterexample to Tachikawa's second conjecture
Over F₂(q,H₁,H₂), with three algebraically independent parameters, the construction gives a finite-dimensional symmetric algebra A and a finite-dimensional nonprojective A-module M with Ext_A^i(M,M)=0 for every i>0.
Sources
- Chen, Fang, and Xi, Tachikawa’s second conjecture, derived recollements, and gendo-symmetric algebras
- OpenAI, A counterexample to Tachikawa’s second conjecture
- A counterexample to Tachikawa's second conjecture
- Tachikawa counterexample manuscript
- Lean Tachikawa theorem
- Later adversarial review and extension
Website timeline
Publication dates and editorial changes.
OpenAI reports a counterexample to Tachikawa’s second conjecture