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On embeddings of rings into their canonical modules

  • Tran Nguyen An
  • Shinya Kumashiro
  • Mouma Samanta
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Abstract

We study embeddings of Cohen--Macaulay local rings into their canonical modules such that the quotient of the cokernel by a regular sequence has the residue field as a direct summand. Motivated by almost Gorenstein rings, we prove an Ext-vanishing criterion for finite projective dimension, which implies G-regularity and the generalized Auslander--Reiten condition. We characterize this summand condition for one-dimensional fiber products and numerical semigroup rings. For Stanley--Reisner rings of graphs, we characterize the corresponding condition for graded embeddings in terms of the graph and show that it is equivalent to a strict multiplicity inequality.

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Cite

BibTeXRIS
BibTeX
@misc{an2026embeddings,
  title = {{On embeddings of rings into their canonical modules}},
  author = {Tran Nguyen An and Shinya Kumashiro and Mouma Samanta},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34395},
  eprint = {2609.34395},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34395},
}
RIS
TY  - UNPB
TI  - On embeddings of rings into their canonical modules
AU  - Tran Nguyen An
AU  - Shinya Kumashiro
AU  - Mouma Samanta
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34395
UR  - https://arxiv.org/abs/2609.34395
AB  - We study embeddings of Cohen--Macaulay local rings into their canonical modules such that the quotient of the cokernel by a regular sequence has the residue field as a direct summand. Motivated by almost Gorenstein rings, we prove an Ext-vanishing criterion for finite projective dimension, which implies G-regularity and the generalized Auslander--Reiten condition. We characterize this summand condition for one-dimensional fiber products and numerical semigroup rings. For Stanley--Reisner rings of graphs, we characterize the corresponding condition for graded embeddings in terms of the graph and show that it is equivalent to a strict multiplicity inequality.
ER  -