{
  "abstract": "Let $(A,\\mathfrak{m})$ 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 $\\mathfrak{m}$-primary ideal $I$ denoted by $\\operatorname{type}_I(A).$ Let $e_i(I)$ denote $i$th Hilbert coefficients of $A$ w.r.t. $I$. Assuming $I$ is integrally closed and $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I) \\neq 0$, we establish a sharp lower bound for $\\operatorname{type}_I(A)$ 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 $e_{2}(\\mathfrak{m})$ using the technique of $S_{2}$-fication. Additionally, for an integrally closed $\\mathfrak{m}$-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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01372",
  "author_total": 2,
  "authors": [
    {
      "name": "Shruti Priya"
    },
    {
      "name": "Samarendra Sahoo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7e19706r5hvdg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01372",
  "fulltext_url": null,
  "id": "7e19706r5hvdg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01372",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7206168949",
  "pdf_url": "https://arxiv.org/pdf/2609.01372",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7e19706r5hvdg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01372",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01372"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Integrally closed ideals with $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)$",
  "title_text": "Integrally closed ideals with e_2(I)=e_1(I)−e_0(I)+λ(A/I)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/7e19706r5hvdg",
  "venue": "arXiv",
  "versions": []
}