No finite repair?

The claim, in plain words

Slice through a point with one function after another, so that nothing collapses at any step: the number of slices is the depth, and a ring whose depth equals its dimension is Cohen–Macaulay. A smooth point is. Two planes in 4-dimensional space touching at a single point are not: dimension 2, depth 1 in our computation. Macaulay’s quartic cone has the same defect, and it can be repaired: its normalization has full depth and is a well-shaped module over it. A nonzero, finitely generated module whose depth equals the dimension of the ring is a small Cohen–Macaulay module.

Hochster asked in the early 1970s whether every complete local domain has one. As the paper tells it, the answer is yes in dimension 2 (the normalization works) and, in positive characteristic, for graded three-dimensional domains over a perfect field (Hartshorne, Peskine–Szpiro, Hochster); big Cohen–Macaulay modules, not necessarily finitely generated, exist in every characteristic. Hochster later came to expect counterexamples; in his 2017 account, “one of these conjectures is bound to be correct”. In September 2026, Anghel argued that cones like this one have no graded such modules, and Liang Chen proposed a counterexample over a completed cone, two days before this paper’s date.

Result 195 is one OpenAI preprint, A Complete Local Domain without a Small Cohen–Macaulay Module (17 pages, dated September 23, 2026). Its Theorem 1.1: there is a three-dimensional complete Noetherian normal local domain R containing ℂ, with residue field ℂ, such that no nonzero finitely generated R-module has depth 3. Corollary 7.1 gives the same for a local ring essentially of finite type over ℂ, a case Bhatt, Hochster and Ma had described as open. The paper says its completed-ring claim overlaps with Chen’s, and that its surface is the construction in Anghel’s earlier preprint.

What it doesn’t claim

Anything in positive characteristic, where graded three-dimensional cones over a perfect field do have such modules, or in mixed characteristic; anything about big Cohen–Macaulay modules, which still exist. The ring is explicit, but the proof is by contradiction. Our examples are small rings computed by us and only illustrate the definitions.

Where it stands

Posted as a preprint

OpenAI, 17 pages, dated September 23, 2026

Peer reviewed

Not yet

Checked in Lean

Not the main claim: OpenAI’s Lean checks only the ring’s properties; not run by us

Last checked October 19, 2026. The preprint has not been revised since it was added to the repository in its initial commit on October 6, 2026. OpenAI published a Lean formalization, but its comparator statement checks only that some ring exists that is three-dimensional, complete, a normal Noetherian local domain containing ℂ with residue field ℂ; as stated, an ordinary power series ring would also satisfy it. The main claim is defined in Lean but not proved, and Bogomolov’s inequality is not formalized. By our count the catalogued part takes 11 files and about 7,800 lines, plus about 37,000 lines of supporting work, excluding Mathlib (source). The Lean files were added on October 7, 2026. The paper is not among those OSVerify reviewed, and we have seen no expert review of it or, as far as we know, of the overlapping preprints.

How the proof goes

1. A cone over a surface. A smooth complex surface X with a positive line bundle H, generated by its sections, gives, by stacking its sections, the ring of a three-dimensional cone; complete it at the tip to get R. Three sections of H give a finite map X → ℙ² of degree H², so R is finite over A = ℂ[[x₀,x₁,x₂]].
2. A module would be free below. A small Cohen–Macaulay module M over R has full depth over the regular ring A, so it is free: M ≅ A^N (eq. (5)).
3. Blow up and balance. Blow up both tips; extend M across the new surface and restrict it there; modify the extension until the slopes of the layers it is built from lie within one unit (Lemma 3.1).
4. A variance squeezed from both sides. Because M is free, a variance v of Chern numbers is at least 0 on the plane side (Lemma 4.1); Bogomolov’s inequality, the Hodge index theorem and Riemann–Roch cap it at 1/2 − (K² − 2c₂)/(12H²) (Lemma 5.1 and the proof of Theorem 1.2). So a module needs K² − 2c₂ ≤ 6H² (Theorem 1.2).
5. A surface that breaks the bound. Blow up the four points of a complete quadrangle and take a cover of degree 7⁵ branched along the six lines and four exceptional curves (the construction in Anghel’s September 2 preprint, re-verified in the paper): H² = 1,715 and K² − 2c₂ = 13,377, more than 6H² = 10,290.

The numbers in the video

• Depth by Koszul homology (ours, exact modulo a prime): smooth 3-space depth 3; two planes touching at a point dimension 2, depth 1; twisted cubic cone 2, 2; Macaulay’s quartic 2, 1, its normalization 2, 2; elliptic cone 2, 2; the cone over E × E dimension 3, depth 2.
• The family of surfaces (ours, from the paper’s formulas, H² = 5n³, K² − 2c₂ = n³(n² − 10)): the bound fails exactly when n ≥ 7; at n = 6 (not prime, so outside the paper’s statement) the formulas give Chen’s constant 99, so Theorem 1.2 does not reach the exponent-six cover Chen uses.
• The upper end of the variance window for n = 7, −0.150, is our arithmetic. The Lean counts are our counts of the import closures.

Read the source

• A Complete Local Domain without a Small Cohen–Macaulay Module (PDF), 17 pages.
• The Lean scope note.
• The collection’s catalog, where this is result 195.

The paper

OpenAI groups related papers into one result; this one has a single paper.

• A Complete Local Domain without a Small Cohen–Macaulay Module

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.