View source for The little finitistic dimension conjecture

'''The little finitistic dimension conjecture''' is the subject of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). For a finite-dimensional algebra ''A'' over a field, the little finitistic dimension is the supremum of the projective dimensions of the finitely generated left ''A''-modules that have finite projective dimension. The conjecture, publicized by Bass in 1960, asserts that this number is finite for every such algebra. The manuscript claims a counterexample. This page summarizes the published claim 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.

== The original result ==
* '''Claim.''' There is a finite-dimensional unital complex algebra ''A'' such that, for every integer ''m'' ≥ 1, there is a finite-dimensional left ''A''-module ''N''<sub>''m''</sub> with 2''m'' − 2 ≤ pd<sub>''A''</sub> ''N''<sub>''m''</sub> < ∞. Hence the little finitistic dimension of ''A'' is infinite, and so is its big finitistic dimension. Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026 ''An algebra of infinite little finitistic dimension''] (23 September 2026), main theorem.
* '''Construction, as summarized in the manuscript.''' The proof has three stages.
** A finitely presented complex group algebra with a central idempotent and an automorphism defines a "selection" functor. It has finite-dimensional modules that survive exactly ''m'' − 1 iterations; the group is motivated by matrix groups of Abels type.
** This process is realized by a fixed bounded bimodule complex over a directed algebra of global dimension at most two.
** The complex is turned into an ordinary bimodule ''X'' over an algebra ''D'' of finite global dimension, and ''A'' is the trivial extension ''D'' ⋉ ''X''. A bar-construction splitting converts the eventual vanishing of iterates into finite projective dimension and a nonzero iterate into a lower bound. The manuscript notes that this splitting also follows from Minamoto and Yamaura's graded-component decomposition for trivial extensions.
* '''Stated consequences.''' By Rickard's theorem, injective left ''A''-modules do not generate the unbounded derived category of left ''A''-modules. Combined with an auxiliary triangular algebra of Cummings, the manuscript obtains a finite-dimensional complex algebra whose little and big finitistic dimensions are infinite on the left and zero on the right (the corollary on extreme left–right asymmetry). The manuscript also refers to the companion manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026 ''A counterexample to Tachikawa's second conjecture''] in the same collection, which obtains characteristic-two algebras of infinite left finitistic dimensions by a different route.
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/198.md lean/docs/198.md] states that the Lean formalization gives a finite-dimensional complex algebra with a finitely generated module of projective dimension at least 2''m'' − 2 and still finite, for every ''m'' ≥ 1, and separately the left–right asymmetry result. The Comparator statements are <code>LittleFinitistic.lean</code> and <code>FinitisticAsymmetry.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-07 (v5: 2026-10-13) || Q. Tang (唐琪越), ''A note of refinements to an algebra of infinite little finitistic dimension'' ([https://hexagonmath.org/2610.00140v5 hexagon:2610.00140v5]); Lean formalization [https://github.com/tqyqiqi1509/Note198-formalization tqyqiqi1509/Note198-formalization] (added 10 October 2026) || Extension || The abstract states that the note simplifies the exposition and extends the construction to every field. The repository README (Lean 4.34.1) reports that 28 Comparator groups covering 58 targets passed with only the standard axioms, and states that the formalization covers the listed targets, not every sentence of the manuscript. || Hexagon, five versions dated 7 to 13 October 2026. The provenance statement describes adversarial review and revision with GPT-6-Astra models and says that the paper is not intended for journal submission.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math at commit <code>adc7f12</code>, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "full": the challenge <code>LittleFinitistic</code> states unbounded finite projective dimensions over a finite-dimensional complex algebra, as in the summary, and <code>FinitisticAsymmetry</code> is a secondary result. It records sandboxed Comparator passes for both challenges (8 October 2026). || The review states that "full" means the Lean states the headline and "does not say the mathematics is correct beyond what Lean checked"; it describes itself as a model-assisted reading without a referee yet.
|-
| 2026-10-14 || A. Jiang, ''Another Counterexample to Little Finitistic Dimension Conjecture'' ([https://hexagonmath.org/2610.00232v1 hexagon:2610.00232v1]) || Variant construction || Gives a characteristic-zero counterexample, described as "arguably providing a simpler variant of OpenAI's construction". The required nilpotence comes from the Rees algebra of ''U''(gl<sub>2</sub>) for its Poincaré–Birkhoff–Witt filtration (Artin–Schelter regularity is cited from Le Bruyn–Smith and Le Bruyn–Smith–Van den Bergh), the graded Beilinson theorem of Minamoto and Mori, and an endofunctor built from tensoring with the dual standard representation and the Casimir element, which lowers the modules attached to Sym<sup>''m''</sup> one step at a time. The transfer to projective dimension adapts the manuscript's square-zero step as a square-zero monad and gives finite-dimensional modules ''L''<sub>''m''</sub> with 2''m'' ≤ pd ''L''<sub>''m''</sub> < ∞ over one finite-dimensional algebra. || Hexagon v1, marked "Primarily AI-generated" with human understanding of "some parts". The provenance statement says that the note was produced in conversation with GPT 6 and that the author has not checked the relevant representation-theory inputs.
|}

== Notes ==
* 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]]