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PreprintCC BY 4.0arXiv

Integrally closed ideals with e2(I)=e1(I)−e0(I)+λ(A/I)

  • Shruti Priya
  • Samarendra Sahoo
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Abstract

Let (A,m) be a Cohen-Macaulay local ring of dimension d. We introduce and study the notion of the generalized type of A with respect to an m-primary ideal I denoted by typeI(A). Let ei(I) denote ith Hilbert coefficients of A w.r.t. I. Assuming I is integrally closed and e2(I)=e1(I)−e0(I)+λ(A/I) ≠ 0, we establish a sharp lower bound for typeI(A) in terms of the multiplicity and certain lengths associated to I. We further show that when this lower bound is attained, the associated graded ring G(I), is Cohen Macaulay. In the case of Buchsbaum local rings of dimension d and depth at least d−1, we obtain an optimal lower bound for e2(m) using the technique of S2-fication. Additionally, for an integrally closed m-primary ideal I, we also study the second extremal case e2(I)=e1(I)−e0(I)+λ(A/I)+1 and its consequences on G(I). We also investigate bounds on e3(I) and for d=3, we study the consequences when these bounds are attained for.

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Cite

BibTeXRIS
BibTeX
@misc{priya2026integrally,
  title = {{Integrally closed ideals with $e_2(I)=e_1(I)-e_0(I)+λ(A/I)$}},
  author = {Shruti Priya and Samarendra Sahoo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01372},
  eprint = {2609.01372},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01372},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Integrally closed ideals with e_2(I)=e_1(I)−e_0(I)+λ(A/I)
AU  - Shruti Priya
AU  - Samarendra Sahoo
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01372
UR  - https://arxiv.org/abs/2609.01372
AB  - Let (A,m) be a Cohen-Macaulay local ring of dimension d. We introduce and study the notion of the generalized type of A with respect to an m-primary ideal I denoted by type_I(A). Let e_i(I) denote ith Hilbert coefficients of A w.r.t. I. Assuming I is integrally closed and e_2(I)=e_1(I)−e_0(I)+λ(A/I) ≠ 0, we establish a sharp lower bound for type_I(A) in terms of the multiplicity and certain lengths associated to I. We further show that when this lower bound is attained, the associated graded ring G(I), is Cohen Macaulay. In the case of Buchsbaum local rings of dimension d and depth at least d−1, we obtain an optimal lower bound for e_2(m) using the technique of S_2-fication. Additionally, for an integrally closed m-primary ideal I, we also study the second extremal case e_2(I)=e_1(I)−e_0(I)+λ(A/I)+1 and its consequences on G(I). We also investigate bounds on e_3(I) and for d=3, we study the consequences when these bounds are attained for.
ER  -