{
  "abstract": "Let $R$ be a commutative ring and let $fPD(R)$ be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if $R$ is Noetherian, the maximal ideals of $R[X]$ detect exactly the Goldman primes of $R$, and this gives the formula $$ fPD(R[X]) =1+\\sup\\{depth R_{\\mathfrak p}\\mid \\mathfrak p\\in GSpec(R)\\}.$$ We construct Noetherian local rings $R_m$ with $fPD(R_m)=0$ and $fPD(R_m[X])=m+1$, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak $(n,d)$-conditions. For every $n\\geq2$ and $h\\geq1$, we construct a local idealization $T$ such that every $n$-presented cyclic $T$-module is projective, while $fPD(T)=h$. The ring $T$ has no nonzero proper finitely presented ideals, but it has a finitely generated ideal $I$ with $Ext_T^i(T/I,T)=0$ for $i<h$ and $Ext_T^h(T/I,T)\\neq0$. Thus the cyclic condition controls $fPD$ at presentation level one, but at no higher level.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14326",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaolei Zhang"
    },
    {
      "name": "Hwankoo Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hpgk2q0ca9rt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14326",
  "fulltext_url": null,
  "id": "0hpgk2q0ca9rt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14326",
  "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": "W7213268657",
  "pdf_url": "https://arxiv.org/pdf/2609.14326",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hpgk2q0ca9rt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14326",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14326"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Goldman primes and higher cyclic presentations for the small finitistic dimension",
  "title_text": "Goldman primes and higher cyclic presentations for the small finitistic dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/0hpgk2q0ca9rt",
  "venue": "arXiv",
  "versions": []
}