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PreprintCC BY 4.0arXiv

Reduction numbers for witnesses to the generalized Loewy length

  • Richard Bartels
  • Sarah Dajani
  • Gabriel Koomson
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Abstract

Let (R,m) be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number rz(md) of md with respect to a witness z∈ md ∖ md+1 to the generalized Loewy length gℓℓ(R) for several infinite families of hypersurfaces {(R,m) }. For every principal reduction w of md, we have gℓℓ(R) ≤ d(rw(md)+1). We give examples of families { (R,m) } such that d(rz(md)+1)−gℓℓ(R)=0 and d(rz(md)+1)−gℓℓ(R)=1. In particular, we prove that for every prime p, there are local hypersurfaces R over 𝔽p such that d(rz(md)+1)−gℓℓ(R)=0 and d(rz(md)+1)−gℓℓ(R)=1. We also prove that the difference d(rz(md)+1)−gℓℓ(R) can vary independently of gℓℓ(R)−e(R).

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Cite

BibTeXRIS
BibTeX
@misc{bartels2026reduction,
  title = {{Reduction numbers for witnesses to the generalized Loewy length}},
  author = {Richard Bartels and Sarah Dajani and Gabriel Koomson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.26003},
  eprint = {2608.26003},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26003},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Reduction numbers for witnesses to the generalized Loewy length
AU  - Richard Bartels
AU  - Sarah Dajani
AU  - Gabriel Koomson
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.26003
UR  - https://arxiv.org/abs/2608.26003
AB  - Let (R,m) be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number r_z(m^d) of m^d with respect to a witness z∈ m^d ∖ m^d+1 to the generalized Loewy length gℓℓ(R) for several infinite families of hypersurfaces {(R,m) }. For every principal reduction w of m^d, we have gℓℓ(R) ≤ d(r_w(m^d)+1). We give examples of families { (R,m) } such that d(r_z(m^d)+1)−gℓℓ(R)=0 and d(r_z(m^d)+1)−gℓℓ(R)=1. In particular, we prove that for every prime p, there are local hypersurfaces R over 𝔽_p such that d(r_z(m^d)+1)−gℓℓ(R)=0 and d(r_z(m^d)+1)−gℓℓ(R)=1. We also prove that the difference d(r_z(m^d)+1)−gℓℓ(R) can vary independently of gℓℓ(R)−e(R).
ER  -