The small Cohen–Macaulay conjecture
A small Cohen–Macaulay module over a local ring is a finitely generated module whose depth equals the dimension of the ring. The small Cohen–Macaulay conjecture. Every complete local domain has a small Cohen–Macaulay module. The conjecture asks for the existence of finitely generated maximal Cohen–Macaulay modules over complete local domains; the paper discusses a new instance of this existence problem and a theorem establishing such a module under additional quasi-Gorenstein hypotheses.
References
Primary source
Likun Xie, “On an Instance of the Small Cohen-Macaulay Conjecture”, arXiv:2302.11011 (2023).
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI. The manuscript claims a three-dimensional complete Noetherian normal local domain containing C, with residue field C, that has no nonzero finitely generated module of depth three. This is a claimed counterexample to the page’s small Cohen–Macaulay module conjecture in its usual nonzero-module convention. It does not exclude infinitely generated big Cohen–Macaulay modules.See full solution
Claimed by OpenAI. The manuscript claims a three-dimensional complete Noetherian normal local domain containing C, with residue field C, that has no nonzero finitely generated module of depth three. This is a claimed counterexample to the page’s small Cohen–Macaulay module conjecture in its usual nonzero-module convention. It does not exclude infinitely generated big Cohen–Macaulay modules.
GitHub repository: https://github.com/openai/math
- OpenAI-195-01-A-Complete-Local-Domain-without-a-Small-Cohen-Macaulay-Module.pdfOpen