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Nonexistence of lim Cohen–Macaulay modules following [Che26] and [Ang26]

Primarily AI-generated textHuman understanding: some partsmath.AC — Commutative Algebramath.AG — Algebraic Geometry

Contributed by Andy Jiang ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Using the inequality stated in [ Che26, Main Claim 1.1], we prove nonexistence of lim Cohen–Macaulay sequences for the completed section rings of smooth projective surfaces over C with an ample globally generated divisor H satisfying KY ≡ 4H and 15H2/8 > χ(OY ). Such a pair is exhibited in [Ang26, Corollary 4.3], with H2 = 1080 and χ(OY ) = 1926. For a complete local domain (R, m) with ℓRHj m(R) < ∞ for 0 ≤ j < dim R, we prove that a lim Cohen–Macaulay sequence in the sense of Bhatt, Hochster, and Ma [ BHM26 ] exists if and only if a lim Γm-Cohen–Macaulay sequence exists (Definition 1.1). These may be different sequences. Chen’s geometric inequality [Che26, Main Claim 1.1] obstructs lim Γm-Cohen–Macaulay sequences over the completed section rings above. The comparison therefore rules out lim Cohen–Macaulay sequences over those rings.

Provenance statement

Proof was produced end-to-end by GPT, by reduction on prior work of [Che26], following [Ang26]. Subsequent messages to the AI were asked to understand the overall strategy. The human did not contribute ideas or check the proof.

References

  1. Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen--Macaulay approximations , Mémoires de la Société Mathématique de France, nouvelle série 38 (1989), 5--37. doi:10.24033/msmf.339 (https://doi.org/10.24033/msmf.339).DOI
  2. Tony J. Puthenpurakal, A short note on the non-negativity of partial Euler characteristics , preprint, 2004. https://arxiv.org/abs/math/0409059.link
  3. Cristian Anghel, Polarized varieties without arithmetically Cohen--Macaulay bundles and section rings without graded maximal Cohen--Macaulay modules in characteristic zero , preprint, arXiv:2609.15589v1, 14 September 2026. https://arxiv.org/abs/2609.15589v1.arXiv
  4. Bhargav Bhatt, Mel Hochster, and Linquan Ma, Lim Cohen--Macaulay sequences of modules , Journal für die reine und angewandte Mathematik 835 (2026), 111--169. doi:10.1515/crelle-2026-0021 (https://doi.org/10.1515/crelle-2026-0021). The numbering used here agrees with the author manuscript dated 10 February 2026 (https://sites.lsa.umich.edu/hochster/wp-content/uploads/sites/1337/2026/02/limCM-Crelle-final.pdf).DOI
  5. Liang Chen, A local cohomology obstruction to small Cohen--Macaulay modules , preprint, arXiv:2609.24142v1, 21 September 2026. https://arxiv.org/abs/2609.24142v1.arXiv
  6. The Stacks Project Authors, The Stacks Project , 2026. https://stacks.math.columbia.edu/. Specific tags are linked in the text. Accessed 22 October 2026.link

Version history

  1. v1Submitted by Andy JiangInitial depositCurrentOct 22, 2026