{
  "abstract": "Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\\mathrm{GL}(\\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash--Miller--Patzt, uses the sharbly resolution. The other, by Brown--Chan--Galatius--Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38735",
  "author_total": 2,
  "authors": [
    {
      "name": "Urshita Pal"
    },
    {
      "name": "Sam Payne"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7fta1af40faj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38735",
  "fulltext_url": null,
  "id": "d7fta1af40faj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38735",
  "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.AT"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7215035387",
  "pdf_url": "https://arxiv.org/pdf/2609.38735",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7fta1af40faj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38735",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38735"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide",
  "title_text": "The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/d7fta1af40faj",
  "venue": "arXiv",
  "versions": []
}