View source for The Auslander–Reiten conjecture

The '''Auslander–Reiten conjecture''' asks whether a finitely generated module ''M'' over an Artin algebra ''R'' must be projective when Ext<sup>''i''</sup>(''M'', ''M'' ⊕ ''R'') = 0 for every ''i'' &gt; 0. Two manuscripts dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claim counterexamples to it and to Tachikawa's second conjecture, with further homological conjectures failing for an associated algebra. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the Auslander–Reiten manuscript:
** The conjecture originates in the work of Maurice Auslander and Idun Reiten on the generalized Nakayama conjecture. Its module and generator formulations are recalled by Huneke and Leuschke.
** The Gorenstein-projective conjecture asks the same question for Gorenstein-projective modules with vanishing self-extensions, as stated by Luo and Huang; it is the restriction of the Auslander–Reiten conjecture to such modules.
** Tachikawa's second conjecture asks whether a finite-dimensional module over a self-injective algebra is projective when all its positive-degree self-extension groups vanish.

== The original results ==
* '''Auslander–Reiten.''' Theorem 1.1 of ''An explicit counterexample to the Auslander–Reiten conjecture'' states that over the field '''F'''<sub>2</sub>(''q'', ''H''<sub>1</sub>, ''H''<sub>2</sub>) with algebraically independent ''q'', ''H''<sub>1</sub>, ''H''<sub>2</sub> there are a finite-dimensional algebra Λ and a finite-dimensional nonprojective Gorenstein-projective module ''Z'' with Ext<sup>''i''</sup>(''Z'', ''Z'') = 0 = Ext<sup>''i''</sup>(''Z'', Λ) for every ''i'' ≥ 1. The algebra has eight simple modules, and the conclusions persist after every field extension, so both conjectures fail also over an algebraically closed field of characteristic two. The manuscript states that the commutative case is not resolved.
* '''Tachikawa.''' ''A counterexample to Tachikawa's second conjecture'' constructs a finite-dimensional symmetric algebra over the same field with a nonprojective module whose positive-degree self-extensions vanish. Its summary states that the associated endomorphism algebra also gives counterexamples to the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture and the Wakamatsu tilting conjecture, and that these conclusions persist after every extension of the field.
* '''Sources.''' At commit <code>1581d4c</code>: [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026 An explicit counterexample to the Auslander–Reiten conjecture] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026 A counterexample to Tachikawa's second conjecture].
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/199.md lean/docs/199.md] states that the Lean formalization gives both counterexamples over '''F'''<sub>2</sub>(''q'', ''H''<sub>1</sub>, ''H''<sub>2</sub>), with the eight simple modules and the persistence under field extension for the Auslander–Reiten example. It states that the later field-extension, endomorphism-algebra and related homological consequences of the Tachikawa manuscript are not included. The Comparator challenges are <code>AuslanderReiten</code> and <code>Tachikawa</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 23 October 2026)
|-
| 2026-10-07 || Q. Tang (唐琪越), ''A note on counterexamples to homological conjectures'' ([https://hexagonmath.org/2610.00143v2 hexagon:2610.00143v2]) || Extension || Building on OpenAI's constructions, extends them to any base field containing an element of infinite multiplicative order. Over each such field it gives finite-dimensional algebras that are counterexamples to both Tachikawa conjectures, the Auslander–Reiten conjecture, the Gorenstein-projective conjecture, the Nakayama, generalized Nakayama and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, the little and big finitistic dimension conjectures, and the Wakamatsu tilting conjecture. || Hexagon, version 2 of 7 October 2026. See also [[The little finitistic dimension conjecture]].
|-
| 2026-10-07 || mio-qwq/math, ''ARC core certificate and finite resonant Yoneda and Tate algebras'' ([https://github.com/mio-qwq/math/tree/029e05b3f45f43690f19dd599834690bcbbf016e/003-arc-one-parameter mio-qwq/math, 003-arc-one-parameter]) || Extension (finite-order variants) || Transcribes the manuscript's ten-dimensional algebra, its trivial extension and its cochain data into a checker, with attribution. Starting from one of the manuscript's two-cone targets over fields of characteristic two, it constructs finite triangular modules for arbitrary nonzero twists and computes their positive Yoneda algebra and, when the twist ratio has finite order, the full Tate cohomology algebra. It states that for finite twist-ratio order the module has nonzero self-extensions and is therefore not a counterexample to the Auslander–Reiten conjecture. || Repository work in an AI-assisted research repository; written proofs with partial Lean formalization; no first-proof, novelty or priority claim.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>bb5fdb6</code>, 18 October 2026) || Review (fidelity of the formal statements) || Rates this family "partial" and records sandboxed Comparator passes for <code>AuslanderReiten</code> and <code>Tachikawa</code>. Its notes state that the consequences of the associated endomorphism algebra named in the family's summary are not among the formal statements. || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-08 (latest commit <code>a371d87</code>, 10 October 2026) || J. Guo, [https://github.com/aster2024/openai-math-followups/blob/a371d874a63222dd662a2ce4ce0562df0c210eb2/consequences/README.md ''Consequences of results in the OpenAI mathematics release''], entry 4, in the GitHub repository aster2024/openai-math-followups || Use (conditional consequence) || Conditional on Theorem 1.1 of the Tachikawa manuscript, combines it with a proposition of Enomoto and Marczinzik to conclude that Tachikawa's first conjecture fails over every field containing '''F'''<sub>2</sub>(''q'', ''H''<sub>1</sub>, ''H''<sub>2</sub>). The entry notes that Tang's note precedes it and covers a larger class of base fields. || GitHub repository.
|-
| 2026-10-09 (latest commit <code>4696381</code>, 10 October 2026) || [https://github.com/KitaKen1/tachikawa-characteristic-zero/tree/46963814be5cc80f781fd78a3c27d3d93d2c2bd3 KitaKen1/tachikawa-characteristic-zero] (GitHub) || Extension (characteristic zero), with a Lean formalization || A Lean 4 counterexample to Tachikawa's second conjecture over the rational numbers: a finite-dimensional symmetric algebra and a finite nonprojective module whose positive self-extension groups vanish. The README also records a negative answer to the Auslander–Reiten conjecture for Artin algebras with the same witness. || GitHub repository with a PDF manuscript.
|-
| 2026-10-10 || cert-machine, openai/math lane K: run 38019899278 for [https://github.com/carlostoledo1891/cert-machine/blob/61aae8eb42f1967e16d37d78cfe8a5c1be6d0d72/corpus/openai-math/kernel-runs/38019899278/AuslanderReiten.json AuslanderReiten] and [https://github.com/carlostoledo1891/cert-machine/blob/61aae8eb42f1967e16d37d78cfe8a5c1be6d0d72/corpus/openai-math/kernel-runs/38019899278/Tachikawa.json Tachikawa] || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept both solutions, in 2,870 s and 829 s. || Repository.
|-
| 2026-10-22 (commit <code>5927828</code>) || H. Enomoto, [https://haruhisa-enomoto.github.io/ai-for-math-slides/ slides of an invited talk at the third AI for Math meeting] (in Japanese), on the author's website || Remark (discussion) || Among recent results obtained with AI in the representation theory of algebras, the slides discuss the collection's counterexample to the Auslander–Reiten conjecture which followed about two weeks after the author's joint paper ([https://arxiv.org/abs/2609.19172 arXiv:2609.19172]) on equivalences among the classical homological conjectures. || Talk slides.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]