{
  "abstract": "We determine the arithmetic rank and cohomological dimension of each generic residual intersection of the ideal of submaximal Pfaffians of an alternating matrix of variables. This ideal fails to be a set-theoretic complete intersection. Our methods yield several structural results: in characteristic zero, we compute the $D$-module length of the top local cohomology module supported at this ideal and show that over the integers, it admits a $\\mathbb{Q}$-vector space structure. In particular, the top local cohomology module has no integer torsion. We also establish the $F$-regularity of the generic residual intersection rings considered here. Further, for each admissible height $s$, we construct $s$-residual intersections of complete intersection ideals and height three Gorenstein ideals that are set-theoretic complete intersections in any characteristic. Finally, we pose a general question of independent interest concerning the behaviour of arithmetic rank.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25680",
  "author_total": 3,
  "authors": [
    {
      "name": "Manav Batavia"
    },
    {
      "name": "Vaibhav Pandey"
    },
    {
      "name": "Uli Walther"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ayrethwrzmdt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25680",
  "fulltext_url": null,
  "id": "1ayrethwrzmdt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25680",
  "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": "W7223032543",
  "pdf_url": "https://arxiv.org/pdf/2610.25680",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ayrethwrzmdt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25680",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25680"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the arithmetic rank and cohomological dimension of some residual intersections",
  "title_text": "On the arithmetic rank and cohomological dimension of some residual intersections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/1ayrethwrzmdt",
  "venue": "arXiv",
  "versions": []
}