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Saturation degree of powers of graded ideals

  • Le Tuan Hoa
  • Hop Dang Nguyen
  • Ngo Viet Trung
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Abstract

Let I be a non-nilpotent graded ideal of a standard graded algebra over a Noetherian commutative ring. We show that the saturation degree sdeg In is asymptotically a constant function or a linear function whose slope is one of the generating degrees of I. Furthermore, we give sufficient (and necessary) conditions for a numerical functions to be the function sdeg In of an ideal I generated by forms of the same degree. Our results may provide a better understanding of the initial behavior of the regularity function reg In.

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Cite

BibTeXRIS
BibTeX
@misc{hoa2026saturation,
  title = {{Saturation degree of powers of graded ideals}},
  author = {Le Tuan Hoa and Hop Dang Nguyen and Ngo Viet Trung},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26532},
  eprint = {2610.26532},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26532},
}
RIS
TY  - UNPB
TI  - Saturation degree of powers of graded ideals
AU  - Le Tuan Hoa
AU  - Hop Dang Nguyen
AU  - Ngo Viet Trung
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26532
UR  - https://arxiv.org/abs/2610.26532
AB  - Let I be a non-nilpotent graded ideal of a standard graded algebra over a Noetherian commutative ring. We show that the saturation degree sdeg I^n is asymptotically a constant function or a linear function whose slope is one of the generating degrees of I. Furthermore, we give sufficient (and necessary) conditions for a numerical functions to be the function sdeg I^n of an ideal I generated by forms of the same degree. Our results may provide a better understanding of the initial behavior of the regularity function reg I^n.
ER  -