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PreprintarXiv licencearXiv

On injective dimension of the conormal module

  • Mohsen Gheibi
  • Kaito Kimura
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Abstract

Let Q be a Gorenstein local ring, let I be a proper ideal of Q, and set R=Q/I. We investigate when finite injective dimension of the conormal module I/I2 forces I to be generated by a regular sequence. We prove this implication for every ideal in the linkage class of a complete intersection. When Q is regular, we also establish it under several numerical conditions in terms of the number of generators of I and the type of R. For normal domain quotients, we show that finite injective dimension of I/I2 forces (ht(I)+1)[ωR]=0 in Cl(R), yielding the desired conclusion when the divisor class group is torsionfree. In the graded setting, we further prove the implication for all monomial ideals in polynomial rings. We also obtain a converse to Kunz's description of the first Koszul homology of an almost complete intersection.

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Cite

BibTeXRIS
BibTeX
@misc{gheibi2026injective,
  title = {{On injective dimension of the conormal module}},
  author = {Mohsen Gheibi and Kaito Kimura},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04031},
  eprint = {2610.04031},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04031},
}
RIS
TY  - UNPB
TI  - On injective dimension of the conormal module
AU  - Mohsen Gheibi
AU  - Kaito Kimura
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.04031
UR  - https://arxiv.org/abs/2610.04031
AB  - Let Q be a Gorenstein local ring, let I be a proper ideal of Q, and set R=Q/I. We investigate when finite injective dimension of the conormal module I/I^2 forces I to be generated by a regular sequence. We prove this implication for every ideal in the linkage class of a complete intersection. When Q is regular, we also establish it under several numerical conditions in terms of the number of generators of I and the type of R. For normal domain quotients, we show that finite injective dimension of I/I^2 forces (ht(I)+1)[ω_R]=0 in Cl(R), yielding the desired conclusion when the divisor class group is torsionfree. In the graded setting, we further prove the implication for all monomial ideals in polynomial rings. We also obtain a converse to Kunz's description of the first Koszul homology of an almost complete intersection.
ER  -