{
  "abstract": "Doi and Takeuchi, and independently Blattner, Cohen and Montgomery, have exhibited a correspondence between cleft extensions of an algebra $U$ by a Hopf algebra $H$ and $H$-measurings on $U$ equipped with a compatible normalized 2-cocycle. Each cleft extension is isomorphic as an algebra, a left $U$-module and a right $H$-comodule to the crossed product algebra constructed from the measuring and the cocycle. Cleft $H$-extensions are a special case of $H$-Hopf-Galois extensions, describing noncommutative principal $H$-bundles with affine total and affine base space. Corings are used to describe noncommutative space covers, allowing a description of possibly nonaffine noncommutative spaces. We propose to describe the total space of a noncommutative $H$-principal bundle by an internal $E$-coring in $H$-comodules where $E$ is an $H$-comodule algebra which is itself $H$-Hopf-Galois. This article focuses on the cleft case. As corings involve bimodules and morphisms of bimodules we extend the Doi--Takeuchi correspondence to bimodule $H$-comodules between cleft extensions on one side and bimodules of $H$-coinvariants equipped with bimodule $H$-measurings compatible with 2-cocycles on the other side. In fact, the correspondences for algebras and for bimodules are parts of a biequivalence of suitable bicategories defined in this article. As a corollary one obtains a correspondence between cleft $H$-extensions of corings and corings equipped with a nonabelian Čech 1-cocycle covalued in $H$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29619",
  "author_total": 1,
  "authors": [
    {
      "name": "Zoran Škoda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7czhyyc1h8qng.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29619",
  "fulltext_url": null,
  "id": "7czhyyc1h8qng",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29619",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7223221939",
  "pdf_url": "https://arxiv.org/pdf/2610.29619",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7czhyyc1h8qng.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29619",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29619"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers",
  "title_text": "Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/7czhyyc1h8qng",
  "venue": "arXiv",
  "versions": []
}