Result 195, Algebra

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.

Disproof or counterexample

The bigger picture

Why it matters

Can a singular algebraic space always be paired with a finitely described algebraic object that behaves better at its singularity? The manuscript reports a three-dimensional counterexample, showing that one proposed way to manage singularities can fail.

What changes?

The manuscript constructs a three-dimensional complete Noetherian normal local domain containing the complex numbers, with complex residue field. A local domain is a ring focused at one point and without zero divisors. It admits no nonzero finitely generated maximal Cohen-Macaulay module, a module generated by finitely many elements whose depth equals the ring's dimension. The summary also reports this failure for a three-dimensional local domain essentially of finite type over the complex numbers.

What does that help mathematicians do?

Depth measures how many successive algebraic constraints a module can accommodate without zero-divisor obstructions. Here the required depth is three. The claimed example rules out a universal strategy of attaching such a finite module to every domain, even under completeness and normality. Researchers therefore cannot assume these modules exist when using them to study local algebra; additional hypotheses or different auxiliary objects are needed.

Are there practical applications?

Its immediate value is foundational: it identifies a limit on finite-module methods for understanding singularities. The finite-type example places that obstruction in the local algebra of complex algebraic varieties, not only in complete rings. It clarifies which existence assumptions require proof before researchers can rely on these methods.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Complete Local Domain without a Small Cohen–Macaulay Module

September 23, 2026 17 pages

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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Cite (BibTeX) 
@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}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.