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A counterexample to Tachikawa's second conjecture
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Abstract
We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over $k=\mathbb F_2(q,H_1,H_2)$ with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- 琪越 唐 (Hexagon contributor), A note on counterexamples to homological conjectures (hexagon:2610.00143, v1 2026-10-07, v4 2026-10-15). Extends the constructions of both family papers to any base field containing an element of infinite multiplicative order, and lists counterexamples to both Tachikawa conjectures, the Auslander–Reiten, Nakayama-type, Auslander–Gorenstein, finitistic-dimension and Wakamatsu tilting conjectures; the page marks the authorship as mixed human/AI and says "Human understanding: no parts". [AI-generated per source] [family-level]
- Kenta Kitamura, tachikawa-characteristic-zero (repository with the manuscript "A characteristic-zero Tachikawa counterexample") (GitHub, October 2026). Its README says it adapts this paper's characteristic-two example to a finite-dimensional symmetric Q-algebra with a nonprojective self-orthogonal module, with a Lean 4 proof; the README discloses ChatGPT/Codex assistance and lists independent review as outstanding. [AI-generated per source]
Comments and discussion
- René Marczinzik, Homological conjectures and problems (personal research page, updated 16 October 2026). Discussing the claimed Tachikawa counterexample and its homological consequences, Marczinzik calls for careful checking and proposes a weakly Gorenstein conjecture over fields whose nonzero elements all have finite multiplicative order, outside the fields used by these constructions. [family-level]
Formalization
- OpenAI math release, Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures: Lean scope (mirrored on mathcommunity/oai-math-revised) (lean/docs/199.md, 2026). Describes the release's Lean formalization of Theorem 1.1 (a symmetric algebra over F_2(q,H_1,H_2) and a nonprojective module M with Ext^i(M,M)=0 for i>0), with comparator statement
Tachikawa.lean; the field-extension, endomorphism-algebra and other homological corollaries are listed as not formalized. [family-level] - KitaKen1 (Kenta Kitamura), Formalize Tachikawa's second conjecture over ℚ and record an external Lean counterexample (formal-conjectures PR #7067, opened 2026-10-10, open). Proposes a Formal Conjectures statement of the rational symmetric case marked
answer(False), with the proof in the external repository above; a reviewer accepted the statement's fidelity but asked for independent verification of the linked proof before approval.
Related prior literature mentioned by others
- René Marczinzik, Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra (arXiv:1608.04212, 2016; revised 2 September 2021). Marczinzik cites this work on weakly Gorenstein and gendo-symmetric algebras in their response to the OpenAI counterexamples as background for the proposed restriction to fields with torsion multiplicative group. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Liang Chen, Big Finitistic Dimensions of Radical-Cube-Zero Algebras (arXiv:2610.00433, 2026-09-30). Constructs a ten-dimensional radical-cube-zero algebra over any field with infinite big finitistic dimension; relevant to Corollary 1.3 (infinite little and big finitistic dimension of Γ_K).
Digests and expositions
- AI Proved What?, No gluing, not projective? Auslander–Reiten and Tachikawa (episode 199, October 2026). This AI-assisted video and transcript outline the seed algebra, cone and twist constructions, symmetric transfer and homological consequences, while distinguishing the stated Lean scope from the later consequences. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 199
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026,
author = {{OpenAI}},
title = {{A counterexample to Tachikawa's second conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/199.md. The Lean statements have not been checked by people against the paper's claims.
Family 199: Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures
Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.
Other papers in this family:
- An explicit counterexample to the Auslander-Reiten conjecture · September 23, 2026