{
  "abstract": "We extend Balmer's tensor-triangular geometry to arbitrary presentably symmetric monoidal $\\infty$-categories $\\mathcal V$ by introducing the atomic smashing frame. This construction is functorial in $\\mathcal V$, recovers the classical Balmer spectrum in the stable compactly-rigidly generated case, and yields a formulation of the telescope conjecture for arbitrary $\\mathcal V$. Using dualizable and locally rigid $\\infty$-categories, we identify smashing ideals with locally rigid localizations and establish recollement results for smashing frames and atomic smashing frames. As a consequence, we obtain a recollement principle for the telescope conjecture. We also establish descent for the telescope conjecture along several natural classes of coverings. As a principal application to chromatic homotopy theory, we analyze the structure of the smashing frame of $\\mathrm{Sp}$. We prove a prime decomposition in terms of the smashing frames of the $p$-local $\\infty$-categories $\\mathrm{Sp}_{(p)}$ and, at each prime, an embedding into the product of the smashing frames of the monochromatic layers $\\mathrm{Sp}_{T(n)}$. In particular, the spatiality of the smashing frame of $\\mathrm{Sp}$ reduces entirely to that of $\\mathrm{Sp}_{T(n)}$. Beyond the stable setting, we characterize the telescope conjecture for $\\infty$-topoi in terms of smashing fields, and for connective module categories and hypercomplete connective sheaves in terms of Pierce-type conditions. Finally, we introduce the Serre smashing frame. Over a connective $\\mathbb{E}_\\infty$-ring $R$, this frame sits between the atomic and usual smashing frames, providing an intermediate structural layer in the study of the telescope conjecture for the connective $R$-module $\\infty$-category.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03375",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiacheng Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30ey1xt243et5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03375",
  "fulltext_url": null,
  "id": "30ey1xt243et5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03375",
  "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.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7208731890",
  "pdf_url": "https://arxiv.org/pdf/2609.03375",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30ey1xt243et5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.03375",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03375"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Generalized Telescope Conjecture",
  "title_text": "Generalized Telescope Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/30ey1xt243et5",
  "venue": "arXiv",
  "versions": []
}