OpenAI · 199-01 · 2026-09-23 · Algebra
An explicit counterexample to the Auslander-Reiten conjecture
preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026
Family 199. Counterexamples to conjectures of Auslander–Reiten, Tachikawa, and Nakayama.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over and a finite-dimensional nonprojective left module Z such that 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 199: Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.