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The Auslander-Reiten conjecture for Artin algebras

Let RR be an Artin algebra (for example a finite-dimensional algebra over a field) and MM a finitely generated RR-module. Auslander and Reiten (1975), in their work on the generalized Nakayama conjecture, conjectured that MM must be projective if ExtRi(M,M⊕R)=0\mathrm{Ext}^i_R(M,M\oplus R)=0 for every i>0i>0. It was proved for many classes (complete intersections, radical cube zero, Igusa-Todorov algebras, rings satisfying Auslander's condition), and Schulz's non-projective module without self-extensions lives over a skew field, outside this setting. The Gorenstein-projective conjecture (Luo-Huang) is its restriction to Gorenstein-projective modules. Is every finitely generated module MM over an Artin algebra with ExtRi(M,M⊕R)=0\mathrm{Ext}^i_R(M,M\oplus R)=0 for all i>0i>0 projective?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Representation theory of algebras; homological conjectures
Posed by
Maurice Auslander and Idun Reiten, On a generalized version of the Nakayama conjecture (Proc. AMS, 1975)
Year posed
1975
Years open
51y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1.1: over k=F2(q,H1,H2)k=\mathbb F_2(q,H_1,H_2) there are a finite-dimensional triangular algebra Λ\Lambda with eight simple modules and rad(Λ)4≠0\mathrm{rad}(\Lambda)^4\ne0, and a finite-dimensional nonprojective Gorenstein-projective module ZZ with Exti(Z,Z)=Exti(Z,Λ)=0\mathrm{Ext}^i(Z,Z)=\mathrm{Ext}^i(Z,\Lambda)=0 for all i≥1i\ge1, persisting under every field extension (so also over an algebraically closed field of characteristic two). This disproves the Auslander-Reiten and Gorenstein-projective conjectures; Z⊕ΛZ\oplus\Lambda also refutes the generator form. It does NOT touch the commutative Noetherian form of the conjecture, and gives no example in characteristic zero.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The two counterexamples share a ten-dimensional starting algebra; each manuscript states it contains its own complete proof.

Verification

No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as posed: an explicit finite-dimensional algebra over k=F2(q,H1,H2)k=\mathbb F_2(q,H_1,H_2) with a nonprojective module ZZ satisfying the vanishing for every i≥1i\ge1. formalization.yaml lists ComparatorChallenges/AuslanderReiten.json, declaration OAI.ArExplicit.Statement.main, file OAI/Algebra/AuslanderReiten/All.lean. The statement was read here: q,H1,H2q,H_1,H_2 algebraically independent over F2\mathbb F_2, and a finite-dimensional algebra with a finite-dimensional module that is not projective, is a cokernel in a totally acyclic complex of finite projectives, has vanishing positive Ext into itself and into the algebra, has semisimple quotient k8k^8 and nonzero fourth radical power, and keeps all of this after base change to every field extension. That is the headline claim, including the Gorenstein-projective form. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.

Sources

Changelog1 change

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