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PreprintCC BY 4.0arXiv

Essential Dimension and Faithful Rank of Finite p-Gerbes

  • Tianzhi Yang
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Abstract

Let p≠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 𝒢/k we show that its prime local version satisfies edk(𝒢;p) ≤ qcdimp(𝒢) ≤ edk(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.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026essential,
  title = {{Essential Dimension and Faithful Rank of Finite p-Gerbes}},
  author = {Tianzhi Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01932},
  eprint = {2609.01932},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01932},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Essential Dimension and Faithful Rank of Finite p-Gerbes
AU  - Tianzhi Yang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01932
UR  - https://arxiv.org/abs/2609.01932
AB  - Let p≠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 𝒢/k we show that its prime local version satisfies ed_k(𝒢;p) ≤ qcdim_p(𝒢) ≤ ed_k(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.
ER  -