Browse › Algebra › 195 A counterexample to the small Cohen–Macaulay module conjecture
A Complete Local Domain without a Small Cohen–Macaulay Module
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Abstract
We construct a three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, that has no nonzero finitely generated maximal Cohen–Macaulay module. This disproves the domain form of the small Cohen–Macaulay module conjecture.
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Notes
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Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- mo271, Wikipedia/SmallCohenMacaulayModules:
exists_isSmallCohenMacaulayis false (google-deepmind/formal-conjectures issue #6912, 2026-10-07). Cites this paper as appearing to settle the formal statement negatively and proposes restating it as a negation; the author asks reviewers to check the claim against the source and notes the comparison was prepared with AI assistance. - maximusrufus, with review by bocowgill, Add reference note: claimed counterexample of exists_isSmallCohenMacaulay (unrefereed; status unchanged) (google-deepmind/formal-conjectures PR #7214, opened 2026-10-12, open at time of collection). Started as a change marking the conjecture as solved but was reduced to a docstring note citing this paper; the reviewer approved the note only, "not certification of the manuscript's proof".
Formalization
- OpenAI math release, Lean scope note for family 195 (lean/docs/195.md). The scope note states that the formalization verifies the existence of a three-dimensional complete Noetherian integrally closed local domain over ℂ with residue field ℂ, but does not prove the absence of small Cohen–Macaulay modules or the essentially finite-type counterexample.
- OpenAI math release, SurfaceConeCandidateSupport.lean (openai/math, lean/ComparatorChallenges). Comparator statement
OAI.SmallCM.candidate_ring_propertiesfor the ring-theoretic properties listed in the scope note. - google-deepmind/formal-conjectures, SmallCohenMacaulayModules.lean (formal-conjectures, accessed 2026-10-21). Lean statement
exists_isSmallCohenMacaulayof the conjecture this paper claims to refute; it was still taggedresearch openwhen collected.
Related prior literature mentioned by others
- Cristian Anghel, Smooth Counterexamples to the Eisenbud–Schreyer–Weyman Ulrich Existence Problem (arXiv:2609.02718v1, 2026-09-02). The AI Proved What? exposition identifies this earlier preprint, also cited in the manuscript, as the source of the surface construction used in the claimed counterexample.
- Cristian Anghel, Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero (arXiv:2609.15589v1, 2026-09-14). The AI Proved What? exposition contrasts this earlier graded-module nonexistence claim, also cited in the manuscript, with OpenAI's claim for arbitrary finite modules over the completed local ring.
- Liang Chen, A local cohomology obstruction to small Cohen–Macaulay modules (arXiv:2609.24142v1, 2026-09-21). The AI Proved What? exposition highlights this earlier proposed completed-local counterexample, also cited in the manuscript, which uses the exponent-six Hirzebruch–Kummer surface rather than the paper's exponent-seven surface.
Possibly relevant literature (found by the agent, not checked)
- Kazuma Shimomoto and Ehsan Tavanfar, On local rings without small Cohen-Macaulay algebras in mixed characteristic (arXiv:2109.12700, 2021; accepted in Math. Res. Lett.). Constructs normal local domains with no small Cohen–Macaulay algebra that still have small Cohen–Macaulay modules, which bears on the paper's remark that an algebra obstruction does not exclude finite modules.
- Likun Xie, On an Instance of the Small Cohen-Macaulay Conjecture (arXiv:2302.11011, 2023; to appear in J. Pure Appl. Algebra). Positive result giving small Cohen–Macaulay modules for certain three-dimensional quasi-Gorenstein deformations, part of the positive side of the problem the paper surveys in its introduction.
- Likun Xie, On an Instance of the Small Cohen-Macaulay Conjecture II (arXiv:2608.10962, v1 2026-08-11, v2 2026-10-16). Proves existence of maximal Cohen–Macaulay modules for local rings of depth d−1 with cyclic deficiency module and constructs section-ring examples, a positive counterpart to the paper's section-ring obstruction.
Digests and expositions
- AI Proved What?, No finite repair? Small Cohen–Macaulay modules (episode 195, October 2026; status checked 2026-10-16). AI-assisted video and written transcript explaining the claimed obstruction and exponent-seven example, comparing the earlier Anghel and Chen preprints, and distinguishing the Lean candidate-ring statement from module nonexistence. [AI-generated per source]
- Open Mathematics Courses / Claude Opus 5.5, Small Cohen–Macaulay modules (four-lesson course, undated). Explicitly AI-written exposition developing the completed cone, elementary modifications, numerical obstruction and Hirzebruch–Kummer construction, with five named surface-theory hypotheses and independent review marked incomplete. [AI-generated per source]
News and blogs
- Hiren Dave / TheNextGenTechInsider, OpenAI Math Repository Shows Mismatch Between Headlines and Lean Proofs (Tech Pulse, 2026-10-12). Uses the small Cohen–Macaulay result to illustrate the distinction between the manuscript's claim and the narrower candidate-ring formalization. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 195
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026,
author = {{OpenAI}},
title = {{A Complete Local Domain without a Small Cohen--Macaulay Module}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf}{OAI:A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/195.md. The Lean statements have not been checked by people against the paper's claims.
Family 195: A counterexample to the small Cohen–Macaulay module conjecture
Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.
This is the only paper in this family.