{
  "abstract": "In [10, page 109] Grothendieck sketches the construction of a complex J * (X) made of commutative pro-algebraic groups, associated to a smooth variety X, and for which each J i (X) is a product of local factors called the local generalized jacobians. The complex J * (X) has the property to compute Zariski cohomology of X with coefficients in algebraic groups and plays the role of a homology of X. The purpose of this note is to recast this construction in the setting of higher algebraic group stacks for the fppf topology instead. For this, we introduce a notion of algebraic homology associated to a scheme over a perfect field which is a universal object computing fppf cohomology with coefficients in group schemes. We endow this algebraic homology with a filtration by dimension of supports, and prove that, when X is smooth of dimension n, it is n-connective with respect to Beilinson's t-structure on the category of filtered objects, and that J * (X) appears as its n-th truncation. In a final part we partially extend our constructions and results over arbitrary bases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Bertrand Toen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btz3ekjbzt7q6.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "btz3ekjbzt7q6",
  "issns": [],
  "landing_url": "https://hal.science/hal-05760320v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-ag"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05760320/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btz3ekjbzt7q6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:39Z",
      "native_id": "hal-05760320",
      "source": "hal",
      "url": "https://hal.science/hal-05760320v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ag",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "GENERALIZED LOCAL JACOBIANS AND COMMUTATIVE GROUP STACKS",
  "title_text": "GENERALIZED LOCAL JACOBIANS AND COMMUTATIVE GROUP STACKS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:39Z",
  "url": "https://research.draconitic.com/paper/btz3ekjbzt7q6",
  "venue": "HAL",
  "versions": []
}