OpenAI · 195-01 · 2026-09-23 · Algebra
A Complete Local Domain without a Small Cohen–Macaulay Module
preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026
Family 195. A counterexample to the small Cohen–Macaulay module conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Constructs a three-dimensional complete Noetherian normal local domain over with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 195: A counterexample to the small Cohen–Macaulay module conjecture. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.