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An explicit counterexample to the Auslander-Reiten conjecture

OpenAI · September 23, 2026

Lean

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Abstract

We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over $k=\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective left module Z such that $\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,Z)=\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,\Lambda)=0$ for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples 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 the Auslander–Reiten, Gorenstein-projective, Tachikawa, Nakayama-type, 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]
  • Weiheng Xia, Global obstruction property and its applications (arXiv:2610.12583, 2026-10-08). Proves positive Auslander–Reiten results via a "global obstruction property"; its introduction cites this paper as "a recent preprint by OpenAI [that] proposes a counterexample to ARC" and states that its results do not depend on that counterexample being valid.

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

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

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Cite

@misc{OAI:An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{An explicit counterexample to the Auslander--Reiten conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf}{OAI:An-explicit-counterexample-to-the-Auslander-Reiten-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: