{
  "abstract": "Let $(A,\\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \\geq 1$. Let $M$ be a MCM $A$-module. The function $n \\rightarrow \\ell(\\text{Tor}^A_j(M, N/I^{n+1}N))$ is of polynomial type and let $r_{I, N}^j(M)$ be its degree. In general we have $r_{I, N}^j(M) \\leq r_{I,N}^2(\\text{Syz}^d_A(k))$ for $j \\geq d + 2$. If $A$ is also dominant in the sense of Takahashi and $M$ is non-free then we show $\\limsup_{j \\rightarrow \\infty} r_{I, N}^j(M) = r_{I,N}^2(\\text{Syz}^d_A(k))$. We prove an analogous result when $V$ is a Cohen-Macaulay module of finite injective dimension and the polynomial type function $n \\rightarrow \\ell(\\text{Ext}_A^j(M, V/I^{n+1}V))$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12120",
  "author_total": 1,
  "authors": [
    {
      "name": "Tony J. Puthenpurakal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dfetbz6rk8dn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12120",
  "fulltext_url": null,
  "id": "9dfetbz6rk8dn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12120",
  "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": "W7221151294",
  "pdf_url": "https://arxiv.org/pdf/2610.12120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dfetbz6rk8dn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12120"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings",
  "title_text": "Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/9dfetbz6rk8dn",
  "venue": "arXiv",
  "versions": []
}