{
  "abstract": "Let $(K,v)$ be a Henselian valued field of rank one and of mixed characteristic $(0,p)$, with value group $Γ$ dense in $\\mathbb{R}$ and residue field $k$. We compute all the graded pieces of the filtration induced by $v$ on the Milnor $\\mathrm{K}$-theory modulo $p$ of $K$, in terms of the exterior powers of the $\\mathbb{F}_p$-vector space $Γ/pΓ$ and of the modules of differential forms of $k$. This extends to the non-discrete setting a classical theorem of Kato concerning complete discretely valued fields. We also prove that, when $(K,v)$ is maximally complete, no class escapes this filtration by accumulating at a limit. As an application, we prove the inequality $\\dim_p(K)\\geq\\dim_{\\mathbb{F}_p}(Γ/pΓ)+\\dim_p(k)$ between Kato's $p$-dimensions, and we show that it becomes an equality as soon as $(K,v)$ is maximally complete.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37961",
  "author_total": 3,
  "authors": [
    {
      "name": "Diego Izquierdo"
    },
    {
      "name": "Giancarlo Lucchini Arteche"
    },
    {
      "name": "Gonzalo Manzano Flores"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6zm11smvz4w4a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37961",
  "fulltext_url": null,
  "id": "6zm11smvz4w4a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37961",
  "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.KT"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6zm11smvz4w4a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the cohomological dimension of Henselian valued fields",
  "title_text": "On the cohomological dimension of Henselian valued fields",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6zm11smvz4w4a",
  "venue": "arXiv",
  "versions": []
}