Hochster’s 1973 small Cohen–Macaulay conjecture
An OpenAI manuscript’s counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel
Awaiting review
Importance: major (4 of 5)The takeaway
OpenAI’s 6 Oct release included a model-written counterexample to the domain form of Hochster’s small Cohen–Macaulay conjecture, a central question of commutative algebra since 1973. A survey by Cristian Anghel, whose September preprints started this line of work, gives a complete proof of it and generalizes it to all dimensions.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 22 October 2026
Your AI and this story
- GPT-6 Astra174 days after its cutoff
- Claude Opus 5.5113 days after its cutoff
- Gemini 4 Argon204 days after its cutoff
- Grok 4.7143 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 113 days before it.
Key facts
- Claim: OpenAI family 195, ‘A counterexample to the small Cohen–Macaulay module conjecture’; per the release notes the manuscript is the output of an OpenAI model with no human author; the catalogue links a Lean scope note (lean/docs/195.md)
- Chain (Anghel’s survey): no Ulrich bundles on certain smooth surfaces (Anghel, arXiv 2609.02718, 2 Sep), answering the Eisenbud–Schreyer–Weyman question of 2003 negatively; extended to all dimensions by Angelo Felice Lopez (2609.06678); no aCM bundles / graded MCM modules (Anghel, 2609.15589); local counterexamples to Hochster’s conjecture if K_Y ≡ 4H (Liang Chen, 2609.24142, ‘proposed’) or σ > 2H² (OpenAI)
- Survey, §11: ‘[OAI] claims a complete proof. Its argument is short … and uses only standard tools. Section 10 gives a complete proof of its main theorem, in every dimension.’ Chen’s argument is ‘a proposed argument circulated for verification’
- New in the survey: the three obstructions become one ‘variance identity’, which explains the factor 2 between the graded bound σ ≤ H² and the local bound σ ≤ 2H²; products of Hesse surfaces with projective spaces give complete local domains with an isolated singularity, in every dimension ≥ 3, without small Cohen–Macaulay modules
- OpenAI’s manuscript credits Anghel’s surface inequality and complete-quadrangle construction and ‘acknowledges that its existential claim overlaps with [Ch]’ (survey §11)
- AI disclosure (survey): ‘ChatGPT and Claude were used for exploratory computations, literature search and editorial assistance, including drafting of the proofs in Sections 9 and 10. The author is solely responsible for the final text.’
- Status (survey): ‘none of the five texts has yet appeared in a refereed publication’
- Open (survey §12): mixed characteristic, where the conjecture was originally motivated by Serre’s positivity of intersection multiplicities; reductions mod p of these rings do have small CM modules
What happened
In three weeks of September 2026 a numerical obstruction for vector bundles on polarized surfaces moved from projective geometry to local commutative algebra. Cristian Anghel showed that some smooth surfaces carry no Ulrich bundles (2 Sep), answering a 2003 question of Eisenbud, Schreyer and Weyman. Angelo Felice Lopez extended this to every dimension. Anghel then ruled out arithmetically Cohen–Macaulay bundles (14 Sep), and Liang Chen proposed a local-cohomology argument against Hochster’s conjecture.
The decisive local step came from a manuscript dated 23 Sep and released on 6 Oct as family 195 of OpenAI’s catalogue. It constructs a complete local domain of dimension 3 with no small Cohen–Macaulay module, under the hypothesis σ(Y) > 2H².
Anghel’s 21 Oct survey (arXiv 2610.300271) calls the OpenAI argument “short” and built on “only standard tools”. It gives a complete proof of the main theorem, generalized to every dimension, and unifies the three obstructions in one identity. He used ChatGPT and Claude to draft those proofs.
No expert reactions beyond the survey were found on 22 Oct (one web search; X not searched).
Why it matters
Hochster’s small Cohen–Macaulay conjecture has been one of the guiding open problems of commutative algebra for five decades. A counterexample in equal characteristic zero changes the field’s expectations. Mixed characteristic, where the conjecture came from, stays open. The result also shows a pattern: human preprints and an AI manuscript built on each other within weeks.
What is disputed or not yet verified
| Verification | Complete proof given in a survey by a named specialist (Anghel, arXiv 2610.30027), not independent of AI drafting; Lean scope note from OpenAI; nothing refereed |
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Sources
5 sources from 2 sites. Numbers match the chips in the text.
5 sources: 5 primary
Primary
- arXiv 2610.30027: Anghel, The recent counterexamples to the Eisenbud–Schreyer–Weyman Ulrich problem and to Hochster’s small Cohen–Macaulay conjecture: a surveyarxiv.org, paper
- OpenAI: A Complete Local Domain without a Small Cohen–Macaulay Module (23 Sep manuscript)github.com, paper
- openai/math CONTENTS.md (family 195)github.com, code
- arXiv 2609.02718: Anghel, Smooth counterexamples to the Eisenbud–Schreyer–Weyman Ulrich existence problemarxiv.org, paper
- arXiv 2609.24142: Liang Chen, A local cohomology obstruction to small Cohen–Macaulay modulesarxiv.org, paper
Changes
- Filed (survey PDF read in full text, OpenAI catalogue entry checked)