{
  "abstract": "We study the $\\mathbb{P}^1$-local motivic homotopy theory under the Nisnevich topology on $Sm_S$. We show that although the homotopy category $\\mathcal{H}^{\\mathbb{P}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable $\\mathbb{P}^1$-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on $\\mathbb{P}^1$. Over perfect fields, we then establish the $\\mathbb{P}^1$-invariance of the higher unstable homotopy group sheaves $\\pi_i^{\\mathbb{P}^1}$ ($i \\geq 1$) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable $\\mathbb{P}^1$-connectivity theorem over fields. As a consequence, we obtain a homotopy $t$-structure on the stable $\\infty$-category of $\\mathbb{P}^1$-invariant Nisnevich spectra, and deduce the strict $\\mathbb{P}^1$-invariance of stable $\\mathbb{P}^1$-homotopy group sheaves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26260",
  "author_total": 1,
  "authors": [
    {
      "name": "Dipankar Maity"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pw7ttvj45vge.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26260",
  "fulltext_url": null,
  "id": "0pw7ttvj45vge",
  "issns": [],
  "landing_url": "https://hal.science/hal-05736606v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
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      "math.AT"
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      "2610|Homotopy and Cohomology in Algebraic Topology"
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  "openalex": "W7213567644",
  "pdf_url": "https://hal.science/hal-05736606/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pw7ttvj45vge.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:27Z",
      "native_id": "hal-05736606",
      "source": "hal",
      "url": "https://hal.science/hal-05736606v1"
    },
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26260",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26260"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$\\mathbb{P}^1$ Homotopy Theory of Schemes",
  "title_text": "ℙ^1 Homotopy Theory of Schemes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/0pw7ttvj45vge",
  "venue": "HAL",
  "versions": []
}