A Complete Local Domain without a Small Cohen–Macaulay Module
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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The recent counterexamples to the Eisenbud-Schreyer-Weyman Ulrich problem (2003) and to Hochster's small Cohen-Macaulay conjecture (1973): a survey, with a generalization to arbitrary dimensionarXiv.orgCristian AnghelOct 21, 2026in the index
In September 2026 four preprints and an AI-generated manuscript moved one numerical obstruction from projective geometry to local algebra. For a smooth complex projective surface $Y$ with an ample, globally generated divisor $H$ such that $H^2 2H^2$ (OpenAI, released 6 October). We recall the…