# A counterexample to finitistic-dimension finiteness

> An algebra with finite resolutions of every length refutes the little finitistic-dimension conjecture. Plain-English explainer, Lean status and sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-198/
Last updated: 2026-10-14

**An algebra with finite resolutions of every length refutes the little finitistic-dimension conjecture.**

**Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Algebra · **Entry:** 198

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Over a fixed finite-dimensional algebra, many modules can be rebuilt from free pieces in finitely many steps. The little finitistic-dimension conjecture said the number of steps needed is bounded for each algebra. This paper proves that false. It builds one complex algebra with modules needing at least 2m − 2 steps, for every m, yet always finitely many.

## At a glance

- **First step:** 1960 (Hyman Bass, “Finitistic dimension and a homological generalization of semi-primary rings”)
- **Significance:** Major. It settles a conjecture publicized by Bass in 1960, with a Lean check of the main statement.
- **Trust:** Formal proof listed
- **Field:** Algebra (1 paper)

## The explanation

Start with a small algebra: numbers a + bx, where x² = 0. It is a two-dimensional space with a multiplication. Take the one-dimensional module k, where x acts as zero. Try to rebuild k from free pieces, which are copies of the algebra itself. Map the algebra onto k by sending a + bx to a. The part that is lost, all the bx, is one-dimensional. It is again a copy of k. So the next step faces the same problem, and the process never stops.

Such a sequence of steps is called a projective resolution. A free piece is the simplest kind of projective module. The projective dimension of a module is the number of steps its resolution needs, when that number is finite. For k above it is infinite. In this small algebra, every module with finite projective dimension turns out to be free already, so the bound is 0.

The little finitistic dimension of an algebra collects only the modules whose resolutions stop. It is the largest projective dimension among finite-dimensional modules where that dimension is finite. Modules like k, with endless resolutions, are left out. The little finitistic-dimension conjecture asserts that this number is finite for every finite-dimensional algebra[1]. Hyman Bass publicized these questions in 1960[2].

Much was known in favour. The conjecture held for monomial algebras and for algebras whose radical cubed is zero[3]. Igusa and Todorov proved it for algebras of representation dimension at most three. Huisgen-Zimmermann had shown that the little and big versions of the dimension can differ[4]. But each of her examples had a finite bound.

This paper gives a negative answer[5]. It builds one finite-dimensional algebra over the complex numbers[6]. For every whole number m, it has a module whose projective dimension is at least 2m − 2 and still finite[6]. So m = 1, 2, 3, 4 give lower bounds 0, 2, 4, 6, and the bounds keep growing. The algebra is chosen once, before m varies[7].

The idea is a countdown. The paper first builds an infinite algebra, the group algebra of a specially chosen group[8]. On it a fixed operation acts on modules. For each m there is a module that survives exactly m − 1 rounds of this operation and vanishes on round m[9]. The paper then copies this countdown into a finite-dimensional algebra. There each round adds steps to a resolution. Vanishing means the resolution stops, and survival keeps it going[10].

The one obstacle is making the copy work the same way for every module at once[11]. The paper builds a single fixed piece of machinery, a complex of bimodules, that imitates the operation uniformly. Relations that first hold only approximately are corrected by an extra layer of data. Then a bound on how long resolutions get over a helper algebra makes the result split cleanly.

There are side results. For this algebra, injective modules fail to generate a certain large category of modules, by a theorem of Rickard[12]. A related algebra has these dimensions infinite on the left and zero on the right[13]. A companion paper gives a separate example in characteristic 2, using another method[14].

How sure is this? A computer proof checker called Lean has checked a formal statement of the main result. It gives a finite-dimensional complex algebra with modules of finite projective dimension at least 2m − 2, for every m. Lean also checks the left–right lopsided example. No specialist review of the paper has been published.

**What it doesn't show.** This does not say anything about modules whose resolutions never stop, which were already allowed.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major. Lean checks the main counterexample and the left–right example; no specialist review is published.

## Words used

- **Module:** Something like a vector space, but with scalars taken from an algebra.
- **Projective resolution:** A step-by-step way of rebuilding a module from the simplest building blocks, projective modules.
- **Projective dimension:** The number of steps a module's shortest projective resolution needs.
- **Little finitistic dimension:** The largest finite projective dimension among the finite-dimensional modules of an algebra.
- **Finite-dimensional algebra:** A finite-dimensional vector space with a compatible multiplication, such as square matrices.
- **Bimodule:** A module with the algebra acting on both the left and the right.

## The original paper

- [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf) - September 23, 2026 · Formal proof listed

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/198.md)

## Quoted lines from the paper (14)

[1] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> The little finitistic-dimension conjecture asserts that this number is finite for every such algebra.

[2] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> The finitistic-dimension questions were publicized by Bass in 1960.

[3] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> Important positive cases for little finitistic finiteness include finite-dimensional monomial algebras, by Green, Kirkman, and Kuzmanovich, and Artin algebras with radical cube zero, by Green and Huisgen-Zimmermann.

[4] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> Huisgen-Zimmermann disproved that equality: for every integer \(n\ge2\), she constructed a finite-dimensional monomial algebra with little finitistic dimension \(n\) and big finitistic dimension \(n+1\).

[5] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> This paper gives a negative resolution of the little finitistic-dimension conjecture.

[6] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> There is a finite-dimensional unital complex algebra \(A\) such that, for every integer \(m\ge1\), there is a finite-dimensional left \(A\)-module \(N_m\) with \[ 2m-2\le\pd_A N_m<\infty. \]

[7] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> The algebra in … is fixed before \(m\) varies.

[8] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> The algebra \(R\) is a group algebra.

[9] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> It admits finite-dimensional modules \(Y_m\) that survive exactly \(m-1\) iterations and vanish on the \(m\)th.

[10] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> Eventual vanishing of \(F^rN\) yields finite projective dimension, while a nonzero \(r\)th iterate forces projective dimension at least \(r\).

[11] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> The principal obstacle is to make this realization uniform in \(Y\).

[12] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> Thus injective left \(A\)-modules do not generate for the algebra of …: its big left finitistic dimension is infinite.

[13] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> Thus the little and big finitistic dimensions of \(\Lambda\) are infinite on the left and zero on the right.

[14] From [An algebra of infinite little finitistic dimension](https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf):

> This is a distinct characteristic-two example; the algebra in Theorem … remains the complex example constructed here.

## Other sources

- [H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the AMS 95 (1960), 466–488](https://doi.org/10.1090/S0002-9947-1960-0157984-8)
- [B. Huisgen-Zimmermann, “Homological domino effects and the first Finitistic Dimension Conjecture”, Inventiones Mathematicae 108 (1992), 369–383](https://doi.org/10.1007/BF02100610)

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