{
  "abstract": "Let $H$ be a nontrivial finite-dimensional nonnegatively graded connected Hopf algebra, with $\\ell=\\max\\{d:H_d\\ne0\\}$, and let $A\\ge0$ be a locally finite $H$-module algebra with finite-dimensional split semisimple, $H$-trivial $A_0$. Under the gap $A_d=0$ for $0<d<\\ell$, the shifted standard cells $\\{q^{-r}Ae_x:0\\le r<\\ell\\}$ form a simple-minded collection. A socle estimate for positive syzygies proves negative-Hom vanishing, while the gap makes the degree-zero endomorphism algebra semisimple. A bounded $t$-structure on the compact derived category with these cells is constructed, with standard cells as its simple heart objects. It is also shown that every compact object has a finite-cell representative and that compact $K_0$ is free on $[Ae_x]$ over $\\mathbb{O}_H=K_0(H\\mbox{-}\\underline{\\mathrm{gmod}})$. The same connected-Hopf hypotheses give a Keller--Nicolás weight structure on the large derived category, without a claim of boundedness or preservation of compact objects. Neither a silting generator nor a bounded weight structure on compacts is forced by the gap. These arguments require neither cocommutativity nor finite representation type of $H$. The $p$-DG case, with $\\mathrm{deg}(\\partial)=2$ and $\\ell=2p-2$, is treated as a specialization. Separately, under the stronger gap condition $A_1=\\cdots=A_\\ell=0$, literal cellularity of every finitely generated graded-projective hopfological module holds. Explicit $p$-DG higher-cycle retracts and algebra-valued traces show why additional grading hypotheses are needed; in particular, dropping the gap can produce nonzero $p$-torsion in the Grothendieck group of compact derived categories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28295",
  "author_total": 1,
  "authors": [
    {
      "name": "You Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bhf69ve0hcqxe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28295",
  "fulltext_url": null,
  "id": "bhf69ve0hcqxe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28295",
  "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",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7214193514",
  "pdf_url": "https://arxiv.org/pdf/2609.28295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bhf69ve0hcqxe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Stable Cellularity in Hopfological Algebra",
  "title_text": "Stable Cellularity in Hopfological Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/bhf69ve0hcqxe",
  "venue": "arXiv",
  "versions": []
}