View source for 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 ([https://github.com/openai/math 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''<sub>''X''</sub><sup>2</sup> − 2''c''<sub>2</sub>(''X'') ≤ 6''H''<sup>2</sup>.
** 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.''' [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026 ''A Complete Local Domain without a Small Cohen–Macaulay Module''] in the openai/math repository (commit <code>1581d4c</code>).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/195.md 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 [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/ComparatorChallenges/SurfaceConeCandidateSupport.lean SurfaceConeCandidateSupport.lean].
* '''Related manuscript in the collection.''' The manuscript on [[Lech's multiplicity conjecture]] cites this counterexample.

== Public follow-up work ==
{| class="wikitable"
! 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'' ([https://arxiv.org/abs/2610.30027 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 '''P'''<sup>2</sup> 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''<sub>''X''</sub><sup>2</sup> − 2''c''<sub>2</sub>(''X''))·''H''<sup>''m''−2</sup> ≤ (''m'' + 4)''H''<sup>''m''</sup>. 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 ==
{| class="wikitable"
! 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'' ([https://arxiv.org/abs/2609.02718 arXiv:2609.02718], v1) || Earlier work (cited by the manuscript) || Shows that Beauville's numerical obstruction to Ulrich bundles on surfaces, ''H''<sup>2</sup> ≥ ''K''<sup>2</sup> − 8''χ''(''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'' ([https://arxiv.org/abs/2609.15589 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'' ([https://arxiv.org/abs/2609.24142 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''<sub>''Y''</sub> numerically equivalent to 4''H'', every nonzero finite reflexive module over the completed vertex local ring satisfies a lower bound on its second local cohomology when 15''H''<sup>2</sup>/8 − ''χ''(''O''<sub>''Y''</sub>) &gt; 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.

[[Category:OpenAI results]]