{
  "schema_version": 1,
  "problem_number": "OWR-15178-020",
  "title": "Arbitrary Local-Cohomology Lengths between Consecutive Ideal Powers",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every field k and every integer i >= 0, we construct a fixed complete regular local ring (R,m) of dimension 2i+2 and a fixed prime complete-intersection ideal I with the following property. Every sequence (a_n) of nonnegative integers is realized by ideals J_n satisfying I^(n+1) contained in J_n contained in I^n and length H^i_m(R/J_n) = a_n. All lower local cohomology vanishes, and all transition maps below degree i+1 are zero. Thus even adjacent-power bounds impose no asymptotic restriction on these lengths for arbitrary decreasing cofinal families. The construction inserts a one-column module presentation into each successive homogeneous layer; its length is computed directly in the top Cech module of a power-series ring. The families need not be multiplicative. The result concerns the unrestricted-family formulation and makes no assertion about ordinary, symbolic, or Frobenius powers. This is an unrefereed preprint prepared with AI assistance; no independent review, formal verification, or absolute priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AC"
  ],
  "keywords": [
    "OWR-15178-020",
    "local cohomology",
    "cofinal ideals",
    "infinitesimal thickening",
    "asymptotic length"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15178-020/",
  "pdf_url": "https://eulersolve.org/papers/owr-15178-020/paper.pdf?v=c6f4425b7e0f",
  "doi": "10.5281/zenodo.23624056",
  "zenodo_record_url": "https://zenodo.org/records/23624056",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For any field k and index i>=0, one fixed complete regular local R of dimension 2i+2 and prime complete-intersection I realize every nonnegative integer sequence a_n as length H^i_m(R/J_n), with I^(n+1) contained in J_n contained in I^n; all lower cohomology and transition maps below i+1 vanish. This theorem concerns arbitrary decreasing cofinal families, which need not be multiplicative. Ordinary, symbolic, Frobenius and multiplicative-family questions are not resolved; the broad source remains partially solved. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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      "sha256": "c6f4425b7e0fad331e3031e3c92b96f42764435b28e04db0fe880bfd5cf88775"
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    "source.zip": {
      "sha256": "c590a054893d137a3ed5b85e7ea003a5278499880f636e9023a11a2d70b2118b"
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    "verification_report.md": {
      "sha256": "e16915cdd724f0313db91926d1b4db5d0f274578335a3f4cb159955b1cc6fe47"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}