{
  "abstract": "Let $U$ be a support $τ$-tilting module over a finite-dimensional algebra $A$, and let $\\mathcal{V}\\subseteq\\operatorname{Fac}U$ be any torsion class. The exact heart $\\mathcal{H}=\\operatorname{Fac}U\\cap\\mathcal{V}^{\\perp}$ has an Ext-progenerator $P$. We prove that $\\operatorname{Hom}_A(P,-)$ realizes $\\mathcal{H}$ as a submodule-closed exact subcategory of $\\operatorname{mod}B$, where $B=\\text{End}_A(P)$, and preserves minimal syzygies. The global relative $k$-delooping levels are independent of $k\\geq1$. Writing $g=\\operatorname{glo}\\text{-}\\operatorname{dell}\\mathcal{H}$, we obtain the sharp comparison $g\\leq\\operatorname{glo}\\text{-}\\operatorname{dell}B\\leq g+1$ and $\\operatorname{findim}B^{\\mathrm{op}}\\leq g+1$. Thus the two global levels are finite simultaneously. For wide hearts the relative levels equal their classical counterparts. We also characterize syzygy finiteness of $B$ by a finite relative syzygy stratum and obtain explicit bounds for both finitistic dimensions. If the heart is Frobenius, it is exactly the category of finitely generated Gorenstein-projective $B$-modules under this realization, and both self-injective dimensions of $B$ are at most one. We also give examples to show the main results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33762",
  "author_total": 3,
  "authors": [
    {
      "name": "Weiqing Cao"
    },
    {
      "name": "Mingfei Xu"
    },
    {
      "name": "Xiaojin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8swpq906tykm3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33762",
  "fulltext_url": null,
  "id": "8swpq906tykm3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33762",
  "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.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7223935819",
  "pdf_url": "https://arxiv.org/pdf/2610.33762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8swpq906tykm3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Relative delooping levels on intervals of torsion classes",
  "title_text": "Relative delooping levels on intervals of torsion classes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8swpq906tykm3",
  "venue": "arXiv",
  "versions": []
}