{
  "abstract": "We develop the chromatic theory of dualizable stable categories, with chromatic purity for continuous $K$-theory at its center. We generalize the chromatic purity theorem for algebraic $K$-theory to continuous $K$-theory of dualizable stable categories and reformulate it as the purity of $H$-unital rings. Using categorical completion theory, we construct chromatic fracture squares and obtain a refinement of chromatic purity: for every dualizable stable category $\\mathcal{C}$, the $T(n)\\oplus T(n-1)$-completion map induces an equivalence $$ K_{T(n)}^{\\mathrm{cont}}(\\mathcal{C})\\xrightarrow{\\simeq}K_{T(n)}^{\\mathrm{cont}}(\\mathrm{Nuc}_{T(n)\\oplus T(n-1)}(\\mathcal{C})). $$ We also establish chromatic descent for continuous $K$-theory of dualizable homotopy fixed points and prove a categorical nuclear refinement of descent. This framework allows us to lift redshift bounds, Tate vanishing, and blueshift from spectra to dualizable stable categories. We show that the $T(n)$-completion of a rigid symmetric monoidal stable category is equivalent to the dualizable limit of module categories over a tower of type $n$ generalized Moore spectra. As applications, we prove that both nuclear solid and nuclear gaseous module categories satisfy chromatic redshift.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26497",
  "author_total": 1,
  "authors": [
    {
      "name": "Yifan Jin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/88wm80xyz7ben.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26497",
  "fulltext_url": null,
  "id": "88wm80xyz7ben",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26497",
  "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.AT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214071475",
  "pdf_url": "https://arxiv.org/pdf/2609.26497",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/88wm80xyz7ben.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26497",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26497"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Chromatic Purity of Dualizable Categories",
  "title_text": "Chromatic Purity of Dualizable Categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/88wm80xyz7ben",
  "venue": "arXiv",
  "versions": []
}