{
  "abstract": "Let X be a primitive double scheme, X = X red , L = I X the ideal sheaf of X, which is a line bundle on X. Let E be a coherent sheaf on X. Then there exists a nonempty open subset U of X and integers m 1 , m 2 , uniquely determined by E, such that E |U is locally isomorphic to\n\n). If we can take U = X, we say the E is quasi locally free. If m 1 = 1 we say that E is rigid. In this case, we show that deformations of E are of the same type. So locally free sheaves and rigid quasi locally free sheaves on X play the same role on X as locally free sheaves do on smooth schemes.\n\nLet E be a quasi locally free sheaf on X. Then E 1 = I X E and E |X are vector bundles on X, and we have a canonical surjective morphism\n\nSimilarly there is a canonical exact sequence Σ(E) of length 4 of vector bundles on X associated to E. Given a surjective morphism Φ : E → E of vector bundles on X, we define an element ∇ E,E (Φ) ∈ Ext 2 X (E, E ⊗ L) such that there exists a quasi locally free sheaf E on X such that Ψ E ≃ Φ if and only ∇ E,E (Φ) = 0, and prove that if E or E can be extended to a vector bundle on X, then E exists. Two ways are given to define ∇ E,E , a direct one and a more complicated one, using the cohomology theory of sheaves of non abelian groups, that can also be used to define an obstruction to the existence of a quasi locally free sheaf E corresponding to a given exact sequence of length 4.\n\nWe study also the existence and properties of flat families of quasi locally free sheaves, giving the possibility to construct universal families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35033",
  "author_total": 1,
  "authors": [
    {
      "name": "Jean-Marc Drézet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fp5zwe3s222cc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35033",
  "fulltext_url": null,
  "id": "fp5zwe3s222cc",
  "issns": [],
  "landing_url": "https://hal.science/hal-05838872v1",
  "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": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223891770",
  "pdf_url": "https://hal.science/hal-05838872/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fp5zwe3s222cc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05838872",
      "source": "hal",
      "url": "https://hal.science/hal-05838872v1"
    },
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35033"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Quasi locally free sheaves on primitive double schemes",
  "title_text": "Quasi locally free sheaves on primitive double schemes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fp5zwe3s222cc",
  "venue": "HAL",
  "versions": []
}