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The finitistic dimension conjecture for finite-dimensional algebras

For a finite-dimensional algebra AA over a field, the little finitistic dimension is findim A=sup⁡{pdAN:N finitely generated, pdAN<∞}\mathrm{findim}\,A=\sup\{\mathrm{pd}_AN: N\text{ finitely generated},\ \mathrm{pd}_AN<\infty\}. The finitistic dimension conjecture, publicized by Bass, asserts that findim A<∞\mathrm{findim}\,A<\infty for every finite-dimensional (more generally, Artin) algebra. It was proved for monomial algebras, radical-cube-zero algebras and representation dimension at most three (Igusa-Todorov), and it implies several other homological conjectures; Huisgen-Zimmermann showed the little and big dimensions can differ but both stayed finite in all examples. Is the little finitistic dimension of every finite-dimensional algebra finite?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Representation theory of finite-dimensional algebras; homological algebra
Posed by
Hyman Bass (publicized the finitistic dimension questions)
Year posed
1960
Years open
66y
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
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a finite-dimensional complex algebra AA with, for each m≥1m\ge1, a finite-dimensional left module NmN_m with 2m−2≤pd Nm<∞2m-2\le\mathrm{pd}\,N_m<\infty; so findim A=∞\mathrm{findim}\,A=\infty and the big finitistic dimension is infinite too, refuting the conjecture for Artin algebras. By Rickard's theorem injective left AA-modules do not generate the derived category. Corollary (left-right asymmetry): an algebra Λ\Lambda with both finitistic dimensions infinite on the left and zero on the right. The construction passes through a finitely presented group algebra with Abels-type central involutions. A companion OpenAI manuscript (Tachikawa family) independently gives characteristic-two examples.

What the AI did

The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture. The proof was not refereed. lean/formalization.yaml lists OAI.LittleFinitistic.Main.exists_counterexample (comparator LittleFinitistic). Its statement ComparatorChallenges/LittleFinitistic.lean was read: there is a finite-dimensional C\mathbb C-algebra AA whose little finitistic dimension (supremum of projective dimensions of finitely generated left modules of finite projective dimension, Mathlib's projectiveDimension) is infinite, with, for every m≥1m\ge1, a finitely generated module NN with 2m−2≤pd N<∞2m-2\le\mathrm{pd}\,N<\infty. This states the headline claim. A second catalogued result, OAI.LittleFinitistic.extreme_left_right_asymmetry (FinitisticAsymmetry), states the corollary on left-right asymmetry and injective generation; read here too. Not rebuilt here.

Sources

Changelog1 change

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