{
  "abstract": "Let $n\\geqslant 1, Ω\\subset\\mathbb{H}^n $ be a domain. We construct a Koszul-type complex for the ideal sheaf $\\mathcal{I}_X^{(k)}$ of $k$-regular functions vanishing on $X=\\{(q_0, q_1, \\cdots, q_{n-1})\\in Ω: q_0=0\\}$ in several quaternionic variables: $$0\\to \\mathcal{R}^{(k+2)}\\xrightarrow{\\widetilde{\\mathscr{L}}^{(k)}} \\mathcal{R}^{(k+1)}\\oplus\\mathcal{R}^{(k+1)}\\xrightarrow{\\mathscr{L}^{(k)}} \\mathcal{I}_X^{(k)}\\to 0,$$ where $k\\geqslant 0$, $\\mathcal{R}^{(k)}$ is the sheaf of $k$-regular functions on $Ω$, $\\widetilde{\\mathscr{L}}^{(k)}=(-L_1^{(k+2)},L_0^{(k+2)})^{T}$, $\\mathscr{L}^{(k)}=(L_0^{(k+1)},L_1^{(k+1)})$, and $L_0^{(k)},L_1^{(k)}$ are multiplication-like operators on $k$-regular functions. This gives the quaternionic analogue of the classical Koszul complex. And we present the long exact sequence in cohomology for the case $Ω\\cap\\{q_0=0\\}=\\emptyset$ with explicit differential connecting maps, by applying the Cauchy-Fueter complex and cohomological methods. As an application, in the special case $n=1, k=1$, the operator pair $(L_0^{(1)}, L_1^{(1)})$ is shown to be surjective if and only if $H^3(Ω, \\mathbb{R})=0$. Furthermore, a cohomological vanishing criterion is given for $H^1(Ω,\\mathcal{I}_X^{(k)})$; under this criterion, every $k$-regular function on $\\{q_0=0\\}\\capΩ$ extends to a $k$-regular function on $Ω$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.10338",
  "author_total": 2,
  "authors": [
    {
      "name": "Yong Li"
    },
    {
      "name": "Yuchen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5cb6115q5wth.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.10338",
  "fulltext_url": null,
  "id": "c5cb6115q5wth",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.10338",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7168255082",
  "pdf_url": "https://arxiv.org/pdf/2607.10338",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5cb6115q5wth.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2607.10338",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.10338"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Koszul complex in quaternionic analysis and its applications",
  "title_text": "A Koszul complex in quaternionic analysis and its applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/c5cb6115q5wth",
  "venue": "arXiv",
  "versions": []
}