{
  "abstract": "Let $p\\neq\\operatorname{char}(k)$. We extend the Karpenko--Merkurjev theorem from classifying stacks of finite $p$-groups to arbitrary finite gerbes whose geometric inertia groups are $p$-groups, without assuming that the gerbe is neutral or that its band is represented by a group scheme over the base field. We prove that the essential dimension at $p$ is exactly the minimum faithful rank obtained after prime-to-$p$ base change, equivalently the faithful rank over a $p$-closure. We also prove a relative form of the theorem for locally full morphisms of finite $p$-gerbes: the relative faithful rank equals the supremum of the essential $p$-dimensions of the fibers. Finally, we introduce the quotient compression dimension, defined using tame quotient singularities with prescribed fundamental gerbe. For every finite $p$-gerbe $\\mathcal{G}/k$ we show that its prime local version satisfies $$ \\mathrm{ed}_k(\\mathcal{G};p) \\leq \\operatorname{qcdim}_p(\\mathcal{G}) \\leq \\mathrm{ed}_k(\\mathcal{G};p)+1. $$ Thus essential dimension at $p$ determines, up to at most one dimension, the smallest quotient singularity realizing the gerbe after prime-to-$p$ localization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01932",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianzhi Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rxa34p2gsr1k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01932",
  "fulltext_url": null,
  "id": "1rxa34p2gsr1k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01932",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7207864835",
  "pdf_url": "https://arxiv.org/pdf/2609.01932",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rxa34p2gsr1k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01932",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01932"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Essential Dimension and Faithful Rank of Finite p-Gerbes",
  "title_text": "Essential Dimension and Faithful Rank of Finite p-Gerbes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/1rxa34p2gsr1k",
  "venue": "arXiv",
  "versions": []
}