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Limits of Amao multiplicities and Epsilon multiplicity

  • Steven Dale Cutkosky
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Abstract

Let R be a d-dimensional Noetherian local ring with maximal ideal mR and let I be an ideal of R. The epsilon multiplicity ε(I) of I is defined by Ulrich and Validashti in \cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, ε(I)= limn→∞ ℓR((In)sat/In)/nd/d!. We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity aε(I). We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.

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Cite

BibTeXRIS
BibTeX
@misc{cutkosky2026limits,
  title = {{Limits of Amao multiplicities and Epsilon multiplicity}},
  author = {Steven Dale Cutkosky},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.12238},
  eprint = {2608.12238},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.12238},
}
RIS
TY  - UNPB
TI  - Limits of Amao multiplicities and Epsilon multiplicity
AU  - Steven Dale Cutkosky
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2608.12238
UR  - https://arxiv.org/abs/2608.12238
AB  - Let R be a d-dimensional Noetherian local ring with maximal ideal m_R and let I be an ideal of R. The epsilon multiplicity ε(I) of I is defined by Ulrich and Validashti in \cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, ε(I)= lim_n→∞ ℓ_R((I^n)^sat/I^n)/n^d/d!. We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity aε(I). We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.
ER  -