1 · The essentials
The question is whether certain algebraic systems always admit a nontrivial, finitely describable companion with especially good internal structure. The conjecture discussed here predicted that such a companion would exist. The manuscript constructs a system for which none exists. Its example is three-dimensional in the mathematical sense and contains the complex numbers. Even with the additional regularity conditions imposed on the example, the desired companion is impossible. This gives a counterexample to the domain form of the small Cohen–Macaulay module conjecture. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine expecting every workshop to admit a compact attachment that makes a particular set of operations work cleanly. The counterexample is like a workshop for which no attachment of the required kind can work—not merely one for which the first few designs fail.
Why it matters
The result rules out a general existence claim: the stated conditions on a domain do not guarantee a nonzero finitely generated maximal Cohen–Macaulay module. The repository summary also reports an example essentially of finite type over the complex numbers, so the reported failure is not restricted to the complete setting.
2 · Going further
The precise question concerns the existence of a nonzero finitely generated maximal Cohen–Macaulay module over a local domain. Such a module has depth equal to the dimension of the ring. The main manuscript constructs a three-dimensional complete Noetherian normal local domain containing , with residue field , for which no such module exists. Thus there is no nonzero finitely generated -module satisfying .
The context supplied by the abstract is the domain form of the small Cohen–Macaulay module conjecture, which this example refutes. The stated approach is a construction of a counterexample; the supplied abstract does not describe the construction or the argument excluding all candidate modules. The main result has a Lean formalization. The repository summary additionally reports a three-dimensional local domain essentially of finite type over with the same nonexistence property, but no companion manuscript or explanation of the relationship between the two constructions is supplied.
Useful notions
- Commutative rings and domains
- Local rings and residue fields
- Finitely generated modules
- Ring dimension and module depth
- Completion and normality
What was known before / what is new
Before
The source identifies the domain form of the small Cohen–Macaulay module conjecture as the existence claim being challenged. It supplies no earlier partial results, historical attribution, or account of previous methods.
New
The main manuscript supplies a complete Noetherian normal local domain of dimension three, containing and having residue field , with no nonzero finitely generated maximal Cohen–Macaulay module. This is a counterexample to the domain form of the conjecture. The repository summary also claims the same nonexistence property for a three-dimensional local domain essentially of finite type over .
Limits and assumptions
The main example has all the stated hypotheses: it is three-dimensional, complete, Noetherian, normal, and local, and it contains with residue field . The exclusion concerns nonzero finitely generated maximal Cohen–Macaulay modules; it does not assert the absence of modules without these restrictions. The main result has a Lean formalization, but the supplied material does not establish that the additional essentially-of-finite-type claim is covered by it. No construction details, companion manuscripts, or remaining open questions are provided.
Mini glossary
- Local domain —
- A commutative ring with no zero divisors and exactly one maximal ideal.
- Finitely generated module —
- A module whose elements can all be expressed using finitely many generators and coefficients from its ring.
- Maximal Cohen–Macaulay module —
- A finitely generated module over a local ring whose depth equals the dimension of that ring.
- Residue field —
- The field obtained by taking a local ring modulo its maximal ideal.
- Noetherian —
- A condition on a ring requiring every ideal to be finitely generated.
- Normal domain —
- A domain that is integrally closed in its field of fractions.
Keywords : Cohen–Macaulay modules · Local domains · Counterexample
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A Complete Local Domain without a Small Cohen–Macaulay Module
23/09/2026We 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.