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. (Lean)

Manuscript

A Complete Local Domain without a Small Cohen–Macaulay Module

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026}

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.

Result 195 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.