{
  "abstract": "Given a prime $p$, a field $F$ of characteristic different from $p$ and a finite $p$-primary Galois module $V$, a Galois cohomology class in $H^n(F,V)$ is called $R$-trivial if it can be connected to $0$ by a class defined over an open subscheme of the projective line. All classes of $H^n(F,V)$ are $R$-trivial if $F$ has cohomological $p$-dimension at most $n$, by the Bloch-Kato conjecture. We show that this bound is sharp: working over fields of iterated Laurent series, we produce, for every $n\\geq 1$, a field $F$ of cohomological $p$-dimension $n+1$ and a Galois module $V$ such that $H^n(F,V)$ contains classes which are not $R$-trivial. The module $V$ is a quotient of the permutation module on the affine hyperplanes of an elementary abelian $p$-group, arising as a quotient of the absolute Galois group of $F$. We also describe $R$-trivial classes as sums of corestrictions of cup products involving a Kummer class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22758",
  "author_total": 1,
  "authors": [
    {
      "name": "Charles De Clercq"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45cr3d1y4qvgd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22758",
  "fulltext_url": null,
  "id": "45cr3d1y4qvgd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22758",
  "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.AG"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7222981987",
  "pdf_url": "https://arxiv.org/pdf/2610.22758",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45cr3d1y4qvgd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22758",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22758"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$R$-trivial Galois cohomology classes and cohomological $p$-dimension",
  "title_text": "R-trivial Galois cohomology classes and cohomological p-dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/45cr3d1y4qvgd",
  "venue": "arXiv",
  "versions": []
}