{
  "abstract": "Let $(R,\\m)$ be a one-dimensional Cohen--Macaulay local ring with a canonical fractional ideal $R\\subseteq K\\subseteq\\overline R$, and let $v$ denote its embedding dimension. Put $σ(R)=\\sup\\{j\\ge0\\mid\\tr(K^j)\\supseteq\\m\\}$. By definition, $R$ is nearly Gorenstein if and only if $σ(R)>0$. We prove that $R$ is almost Gorenstein if and only if $σ(R)\\ge v$, equivalently, $σ(R)=\\infty$. Thus every finite value of $σ(R)$ is at most $v-1$. We then determine the structure of rings for which $σ(R)$ attains its largest finite value $v-1$, just below the threshold for the almost Gorenstein property. This occurs if and only if $K^{v-1}\\cong\\m$ and $K^v\\not\\cong\\m$. When these conditions hold, $R$ has type two, with multiplicity $e$ greater than $v$ and $e \\equiv 3 \\pmod{v-1}$. As an application, we characterize the numerical semigroup rings $R$ with $σ(R)=v-1$ as those generated by an arithmetic progression, of type two and multiplicity greater than $v$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05150",
  "author_total": 2,
  "authors": [
    {
      "name": "Le Truong Hoang"
    },
    {
      "name": "Naoyuki Matsuoka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7wxfqjmm9yhv0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05150",
  "fulltext_url": null,
  "id": "7wxfqjmm9yhv0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05150",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7220671350",
  "pdf_url": "https://arxiv.org/pdf/2610.05150",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7wxfqjmm9yhv0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05150",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05150"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Traces of powers of canonical ideals",
  "title_text": "Traces of powers of canonical ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7wxfqjmm9yhv0",
  "venue": "arXiv",
  "versions": []
}