{
  "abstract": "Motivated by lifting problems for principal bundles, we study the following algebraic problem. Let $\\smash{\\widehat{G}}$ be a group with identity element $e$, let $G \\leq \\smash{\\widehat{G}}$, and let $S$ be a strongly $G$-graded unital ring with principal component $R:=S_e$. We ask whether the given grading extends to a strong $\\smash{\\widehat{G}}$-grading without altering its $G$-homogeneous components. For every extension $\\smash{\\widehat{p}}$ of the Picard homomorphism of $S$, we construct a characteristic class in the relative third cohomology group \\[ H^3_{\\smash{\\widehat{p}}}(\\smash{\\widehat{G}},G;\\operatorname{U}(Z(R))), \\] where $Z(R)$ denotes the center of $R$ and $\\operatorname{U}(Z(R))$ its group of units. The~vanishing of this class is equivalent to the existence of a lift. When it vanishes, the relative second cohomology group \\[ H^2_{\\smash{\\widehat{p}}}(\\smash{\\widehat{G}},G;\\operatorname{U}(Z(R))) \\] acts simply transitively on the equivalence classes of lifts with Picard homomorphism $\\smash{\\widehat{p}}$. We also prove that an $H$-grading is strong if and only if both its restriction to a normal subgroup $N \\trianglelefteq H$ and its induced $H/N$-grading are strong. We illustrate the theory through a range of examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09879",
  "author_total": 2,
  "authors": [
    {
      "name": "Emma Husen"
    },
    {
      "name": "Stefan Wagner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4hmnpsesv6kz7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09879",
  "fulltext_url": null,
  "id": "4hmnpsesv6kz7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09879",
  "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.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7212190940",
  "pdf_url": "https://arxiv.org/pdf/2609.09879",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4hmnpsesv6kz7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.09879",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09879"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lifting strongly graded rings via overgroups",
  "title_text": "Lifting strongly graded rings via overgroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/4hmnpsesv6kz7",
  "venue": "arXiv",
  "versions": []
}