{
  "abstract": "The (naive) de Rham cohomology of a zero-dimensional scheme is the homology of the Kähler differential algebra of its coordinate ring, viewed as a complex. Since it is well-known to vanish in higher degrees if the ring is quasi-homogeneous, we concentrate on the affine case. For a general affine $K$-algebra $R=P/I$, where ${\\rm char}(K)=0$ and $P=K[x_1,\\dots,x_n]$, we prove a vanishing theorem for $H_{\\rm dR}^m(R)$ based on the shape of a Macaulay basis of $dI\\wedge Ω^m_{P/K}$. For an Artinian local algebra $A=P/\\langle f_1,\\dots,f_n\\rangle$, where $\\{f_1,\\dots,f_n\\}$ is a super regular sequence, we show that $H_{\\rm dR}^{\\bullet}(A)$ is non-trivial in general, but trivial when the natural system of generators of the relation module of $Ω^m_{A/K}$ is a standard basis. Moreover, we provide a detailed study of the dimension of $H_{\\rm dR}^0(A)=\\ker(d_A)$ for Artinian local rings $A=P/I$. The case of arbitrary affine zero-dimensional schemes is reduced to this case using a Galois splitting, and as a result we obtain the de Rham cohomology of a fat point scheme. Many explicitly computed examples and counterexamples support the results and indicate how subtle the de Rham cohomology of a zero-dimensional scheme is in general.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34819",
  "author_total": 2,
  "authors": [
    {
      "name": "Martin Kreuzer"
    },
    {
      "name": "Le Ngoc Long"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/34ct2fn26dev1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34819",
  "fulltext_url": null,
  "id": "34ct2fn26dev1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34819",
  "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"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214728417",
  "pdf_url": "https://arxiv.org/pdf/2609.34819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/34ct2fn26dev1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the De Rham Cohomology of Zero-Dimensional Schemes",
  "title_text": "On the De Rham Cohomology of Zero-Dimensional Schemes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/34ct2fn26dev1",
  "venue": "arXiv",
  "versions": []
}