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Hochster's small Cohen-Macaulay module conjecture for complete local domains

A small (maximal) Cohen-Macaulay module over a Noetherian local ring (R,m)(R,\mathfrak m) is a nonzero finitely generated module MM with depth M=dim⁡R\mathrm{depth}\,M=\dim R. Hochster conjectured in the early 1970s that every complete Noetherian local domain has one. It holds in dimension at most two, and in some graded positive-characteristic cases in dimension three; big Cohen-Macaulay modules and algebras exist in all characteristics, but they are not finitely generated, and the existence question was open even for local rings of affine three-dimensional domains over algebraically closed fields. Does every complete Noetherian local domain admit a small Cohen-Macaulay module?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Commutative algebra; homological conjectures
Posed by
Melvin Hochster
Year posed
1973
Years open
53y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a three-dimensional complete Noetherian normal local domain R⊃CR\supset\mathbb C with residue field C\mathbb C having no small Cohen-Macaulay module; the uncompleted vertex localization of the same section ring, essentially of finite type over C\mathbb C, also has none. The mechanism is Theorem 1.2: if the completed cone R(X,H)R(X,H) over a smooth projective surface has a small Cohen-Macaulay module, then KX2−2c2(X)≤6H2K_X^2-2c_2(X)\le6H^2, applied to a Hirzebruch-Kummer surface with KX≡5HK_X\equiv5H. It does not treat positive or mixed characteristic, the non-domain forms, or dimension three with other residue fields.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, 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). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it gives a three-dimensional complete Noetherian normal local domain containing C\mathbb C with residue field C\mathbb C and no nonzero finitely generated module of depth three, which contradicts the domain form as posed. The obstruction (Theorem 1.2) is KX2−2c2(X)≤6H2K_X^2-2c_2(X)\le6H^2 for completed section rings of surfaces, and the example uses a Hirzebruch-Kummer surface from Anghel's 2026 preprint, with the numerical checks redone in the paper. A reader should know that the manuscript itself reports overlapping recent work: Anghel (graded case, September 14, 2026) and Chen (arXiv 2609.24142, September 21, 2026), who proposes a completed-local counterexample under a proportionality hypothesis. Not refereed. No Lean formalization in the release.

Sources

Changelog1 change

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