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On an Instance of the Small Cohen-Macaulay Conjecture II
Abstract
We show that any d-dimensional local ring A with a dualizing complex, depth A=d−1, and cyclic deficiency module Kd−1(A) admits a maximal Cohen--Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection A→ Kd−1(A). This gives new cases of the small Cohen--Macaulay conjecture. When A is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module ωA/xA, for any x∈annA Kd−1(A) that is regular on A. This recovers a theorem of Tavanfar and Shimomoto for quasi-Gorenstein deformations of a 3-dimensional quasi-Gorenstein local ring A with K2(A)≅ k. We also construct examples of section rings satisfying the hypotheses of our theorem.
Related
- 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
- A local cohomology obstruction to small Cohen--Macaulay modules
- Monomial ideals whose all matching powers are Cohen–Macaulay
- Prime and Cohen-Macaulay ideals of 2-minors with linear resolution
- Adjoint test modules along Cohen--Macaulay morphisms
BibTeX
@misc{xie2026instance,
title = {{On an Instance of the Small Cohen-Macaulay Conjecture II}},
author = {Likun Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.10962},
eprint = {2608.10962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.10962},
}
RIS
TY - UNPB
TI - On an Instance of the Small Cohen-Macaulay Conjecture II
AU - Likun Xie
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2608.10962
UR - https://arxiv.org/abs/2608.10962
AB - We show that any d-dimensional local ring A with a dualizing complex, depth A=d−1, and cyclic deficiency module K^d−1(A) admits a maximal Cohen--Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection A→ K^d−1(A). This gives new cases of the small Cohen--Macaulay conjecture. When A is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module ω_A/xA, for any x∈ann_A K^d−1(A) that is regular on A. This recovers a theorem of Tavanfar and Shimomoto for quasi-Gorenstein deformations of a 3-dimensional quasi-Gorenstein local ring A with K^2(A)≅ k. We also construct examples of section rings satisfying the hypotheses of our theorem.
ER -