Result 198, Algebra

A counterexample to finitistic-dimension finiteness

Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

The manuscript reports that a finite-dimensional algebra can support linear actions of arbitrarily high complexity, even when each has finite complexity. This challenges a conjecture that one algebra should impose a common ceiling on that complexity.

What changes?

An algebra here is a finite-dimensional complex vector space with multiplication. Its finite-dimensional modules are finite-dimensional spaces on which it acts. A module's projective dimension measures the shortest length of a resolution expressing it through special building blocks called projective modules. The manuscript claims to construct one such algebra whose modules have unbounded finite projective dimensions. Their supremum, called the little finitistic dimension, is therefore infinite, contrary to the little finitistic-dimension conjecture.

What does that help mathematicians do?

The crucial point is that the algebra stays fixed while the finite projective dimensions grow without bound. Modules with infinite projective dimension alone would not contradict this conjecture, because they are excluded from the supremum. The claimed example would rule out a universal finiteness statement and show that any valid bound must rely on additional restrictions, not merely the algebra's finite-dimensionality.

Are there practical applications?

The immediate value is foundational: it would clarify the limits of using projective resolutions to measure the complexity of finite-dimensional representations. Researchers seeking finiteness guarantees would need to identify assumptions that exclude this behavior. The supplied abstract describes a mathematical counterexample, not a computational method or practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

An algebra of infinite little finitistic dimension

September 23, 2026 24 pages Main result formalized in Lean

We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-algebra-of-infinite-little-finitistic-dimension-September-23-2026,
  author = {{OpenAI}},
  title = {{An algebra of infinite little finitistic dimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf}{OAI:An-algebra-of-infinite-little-finitistic-dimension-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/198.md.

A counterexample to finitistic-dimension finiteness

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The little finitistic-dimension conjecture predicts that finitely generated modules of finite projective dimension over a fixed finite-dimensional algebra have bounded projective dimensions. The formalized counterexample is a finite-dimensional complex algebra with a finitely generated module of projective dimension at least 2m−22m-2 and still finite for every m≥1m\ge1.

A separate formalized result gives a finite-dimensional complex algebra whose little and big finitistic dimensions are infinite on the left and zero on the right. Injectives fail to generate its unbounded derived category on the left and generate it on the right.

Comparator links

Result Comparator statement
Infinite little finitistic dimension LittleFinitistic.lean
Left/right finitistic-dimension asymmetry FinitisticAsymmetry.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.