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Tachikawa's second conjecture for self-injective algebras

Claims Theorem 1.1: over $k=\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional symmetric algebra $A$ and a nonprojective finite-dimensional module $M$ with $\mathrm{Ext}^i_A(M,M)=0$ for all $i>0$ , so Tachikawa's second conjecture fails, and $M$ 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.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Tachikawa's second conjecture for self-injective algebras · Tachikawa's second conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Tachikawa's second conjecture for self-injective algebras

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No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture: a finite-dimensional symmetric (hence self-injective) algebra over $\mathbb F_2(q,H_1,H_2)$ 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 $\mathbb F_2[X_0,X_1,X_2]$ 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.