{
  "abstract": "This work revisits the key sufficient conditions on a ring $A$ that guarantee the existence of the exact sequence \\[ H_3(\\mathrm{SM}_2(A), \\mathbb{Z})\\to H_3(\\mathrm{SL}_2(A), \\mathbb{Z}) \\to \\mathcal{RB}(A) \\to 0 \\] and of the isomorphism \\[ H_3(\\mathrm{SL}_2(A), \\mathrm{SM}_2(A); \\mathbb{Z}) \\simeq \\mathcal{RP}_1(A). \\] We show that there are rings which satisfy the relations above but do not satisfy the condition that -1 is an square, for example some rings $\\mathbb{Z}[\\frac{1}{m}]$ and finite local rings. In addition, we study the special case of non-dyadic local fields with characteristic not 2, obtaining a refined Bloch-Wigner exact sequence when -1 is not an square.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.08890",
  "author_total": 1,
  "authors": [
    {
      "name": "Elvis Torres Pérez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5exfpz3wwqgkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.08890",
  "fulltext_url": null,
  "id": "5exfpz3wwqgkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.08890",
  "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.KT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7202165836",
  "pdf_url": "https://arxiv.org/pdf/2608.08890",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5exfpz3wwqgkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.08890",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.08890"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Remarks on refined scissors congruence group and the third homology of $\\mathrm{SL}_2$",
  "title_text": "Remarks on refined scissors congruence group and the third homology of SL_2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/5exfpz3wwqgkv",
  "venue": "arXiv",
  "versions": []
}