Hochster's small Cohen–Macaulay module conjecture

Hochster's small Cohen–Macaulay module conjecture, in its domain form, asks whether every complete Noetherian local domain has a small Cohen–Macaulay module: a nonzero finitely generated module whose depth equals the dimension of the ring. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a three-dimensional counterexample over the complex numbers. This page summarizes that claim and public work related to it.

Background

As recounted in the manuscript's introduction:
- Hochster formulated the finite-module existence problem in his work of 1973 and 1975. His later account records both the existence conjecture and a subsequent expectation of counterexamples.
- A complete local domain of dimension at most two has a small Cohen–Macaulay module, since its finite normalization is Cohen–Macaulay.
- By a theorem attributed to Hartshorne, Peskine–Szpiro and Hochster, a three-dimensional finitely generated nonnegatively graded domain whose degree-zero part is a perfect field of positive characteristic has a graded small Cohen–Macaulay module.
- Big Cohen–Macaulay modules, which need not be finitely generated, exist for all Noetherian local rings: Hochster proved this in equal characteristic, and perfectoid methods (André) completed the mixed-characteristic case.
- Bhatt constructed complete normal local domains of positive characteristic with no finite Cohen–Macaulay extension. A version of a paper of Bhatt, Hochster and Ma dated 25 August 2026 describes the existence problem as open even for local rings at maximal ideals of affine three-dimensional domains over algebraically closed fields.

The original result

Claim. Theorem 1.1: there is a three-dimensional complete Noetherian normal local domain R containing C, with residue field C, such that no nonzero finitely generated R-module has depth three. A corollary states that the uncompleted localization at the vertex of the same section ring, a three-dimensional local domain essentially of finite type over C, also has no small Cohen–Macaulay module.
Method, as described in the manuscript. For a smooth integral projective surface X over C and an ample globally generated divisor H, let R(X, H) be the completion of the section ring at its vertex.
- Theorem 1.2: if R(X, H) has a small Cohen–Macaulay module, then K_X² − 2c₂(X) ≤ 6H².
- The proof compares coherent extensions across the exceptional divisor of the completed cone. It keeps the module's vector bundle on the punctured spectrum, so it applies to modules without a grading.
- The example is the complete-quadrangle Hirzebruch–Kummer surface from Anghel's work; the manuscript gives its construction and the required numerical and positivity checks directly.
Source. A Complete Local Domain without a Small Cohen–Macaulay Module in the openai/math repository (commit 1581d4c).
Formalization, as described by the repository. The scope note lean/docs/195.md states that the formalization verifies only the ring-theoretic candidate: a three-dimensional complete Noetherian integrally closed local domain over C with residue field C. "It does not prove that this ring lacks a nonzero finitely generated maximal Cohen–Macaulay module, or the paper's counterexample essentially of finite type." The Comparator challenge is SurfaceConeCandidateSupport.lean.
Related manuscript in the collection. The manuscript on Lech's multiplicity conjecture cites this counterexample.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 22 October 2026)
2026-10-21	C. Anghel, 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 dimension (arXiv:2610.30027, v1)	Survey; extension (every dimension)	Surveys how one numerical obstruction passes from Ulrich bundles to arithmetically Cohen–Macaulay bundles and to small Cohen–Macaulay modules, citing the related work below and this manuscript. It states that, through a finite projection to P² and an identity already present in the manuscript, the three obstructions become one variance identity, which explains the factor 2 between the graded and the local bounds. It generalizes the manuscript's argument to every dimension m ≥ 2: for a smooth complex projective variety X of dimension m and H ample and globally generated, a small Cohen–Macaulay module over the completed section ring forces (K_X² − 2c₂(X))·H^m−2 ≤ (m + 4)H^m. It concludes that products of Hesse surfaces with projective spaces give complete local domains with an isolated singularity, in every dimension from 3 on, without small Cohen–Macaulay modules.	arXiv preprint, version 1; the generalization is stated with complete proofs.

Related work by others

Date	Work	Relation	Public claim	Public status (checked 22 October 2026)
2026-09-02	C. Anghel, Smooth counterexamples to the Eisenbud–Schreyer–Weyman Ulrich existence problem (arXiv:2609.02718, v1)	Earlier work (cited by the manuscript)	Shows that Beauville's numerical obstruction to Ulrich bundles on surfaces, H² ≥ K² − 8χ(O), holds without his Picard-rank-one hypothesis, and exhibits a very ample class on the exponent-3 Hesse surface violating it, a smooth counterexample to the Eisenbud–Schreyer–Weyman question. The manuscript cites the surface inequality and the Hirzebruch–Kummer construction used for its example.	arXiv preprint.
2026-09-14	C. Anghel, Polarized varieties without arithmetically Cohen–Macaulay bundles and section rings without graded maximal Cohen–Macaulay modules in characteristic zero (arXiv:2609.15589, v1)	Earlier work (cited by the manuscript)	Exhibits smooth polarized complex surfaces with no nonzero arithmetically Cohen–Macaulay bundle, so that their section rings, three-dimensional normal graded domains, have no nonzero finitely generated graded maximal Cohen–Macaulay module; it concludes that in characteristic zero Hochster's conjecture admits no graded refinement. The manuscript notes that this excludes graded modules over the same complete-quadrangle section rings that it uses.	arXiv preprint.
2026-09-21	L. Chen, A local cohomology obstruction to small Cohen–Macaulay modules (arXiv:2609.24142, v1)	Earlier proposed obstruction (cited by the manuscript)	Proposes that, for a smooth complex projective surface Y with an ample globally generated H and K_Y numerically equivalent to 4H, every nonzero finite reflexive module over the completed vertex local ring satisfies a lower bound on its second local cohomology when 15H²/8 − χ(O_Y) > 0, which would exclude small Cohen–Macaulay modules.	arXiv preprint; the abstract presents its main statement as a claim. Anghel's survey above lists it as proposed.

Notes

The list covers public work found up to the date above; it is not a priority or completeness claim.