{
  "abstract": "Let $M$ be a finitely generated module over a ring $R$ that is a finitely generated algebra over a Noetherian ring $A$ by $r$ elements. We prove that there are finitely many primes $P_1, \\ldots, P_k$ of $A$ such that $M$ has a countable ascending filtration by finitely generated $A$-modules with the property that every factor $F_n$ has a filtration by cyclic modules of the form $A/P_i$, $1 \\leq i \\leq k$ such that the number of copies $H_i(n)$ of $A/P_i$ that occur in the filtration of $F_n$ is an eventually polynomial function of $n$. We prove a number of bounds on the degrees of these eventually polynomial functions: $r-1$ is always a bound, and the dimension of $\\dim(M_P)-1$ is a bound under mild hypotheses. One application is an immediate proof Grothedieck's classic theorem on generic freeness. Several other applications are given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41200",
  "author_total": 2,
  "authors": [
    {
      "name": "Mel Hochster"
    },
    {
      "name": "Yuanyuan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cne3qmr9pzjkm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41200",
  "fulltext_url": null,
  "id": "cne3qmr9pzjkm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41200",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41200",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cne3qmr9pzjkm.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41200",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41200"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Filtration theorems, Hilbert functions, and generic freeness",
  "title_text": "Filtration theorems, Hilbert functions, and generic freeness",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/cne3qmr9pzjkm",
  "venue": "arXiv",
  "versions": []
}