{
  "id": "2408.09990",
  "version": "v1",
  "published": "2024-08-19T13:41:19.000Z",
  "updated": "2024-08-19T13:41:19.000Z",
  "title": "Hypercomplete étale framed motives and comparison of stable homotopy groups of motivic spectra and étale realizations over a field",
  "authors": [
    "Andrei Druzhinin",
    "Ola Sande"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "For any base field and integer $l$ invertible in $k$, we prove that $\\Omega^\\infty_{\\mathbb{G}_m}$ and $\\Omega^\\infty_{\\mathbb{P}^1}$ commute with hyper \\'etale sheafification $L_{\\acute{e}t}$ and Betti realization through infinite loop space theory in motivic homotopy theory. The central subject of this article is an $l$-complete hypercomplete \\'etale analog of the framed motives theory developed by Garkusha and Panin. Using Bachman's hypercomplete \\'etale \\RigidityTheorem and the $\\infty$-categorical approach of framed motivic spaces by Elmanto, Hoyois, Khan, Sosnilo, Yakerson, we prove the recognition principle and the framed motives formula for the composite functor \\[\\Delta^\\mathrm{op}\\mathrm{Sm}_k\\to \\mathrm{Spt}^{\\mathbb{G}_m^{-1}}_{\\mathbb{A}^1,\\acute{e}t}(\\mathrm{Sm}_k)\\xrightarrow{\\Omega^\\infty_{\\mathbb{G}_m}} \\mathrm{Spt}_{\\acute{e}t,\\hat{n}}(\\mathrm{Sm}_k).\\] The first applications include the hypercomplete \\'etale stable motivic connectivity theorem and an \\'etale local isomorphism \\[\\pi^{\\mathbb{A}^1,\\mathrm{Nis}}_{i,j}(E)\\simeq\\pi^{\\mathbb{A}^1,\\acute{e}t}_{i,j}(E)\\] for any $l$-complete effective motivic spectra $E$, and $j\\geq 0$. Furthermore, we obtain a new proof for Levine's comparison isomorphism over $\\mathbb C$, $\\pi_{i,0}^{\\mathbb{A}^1,\\mathrm{Nis}}(E)(\\mathbb{C})\\cong \\pi_i(Be(E))$, and Zargar's generalization for algebraically closed fields, that applies to an arbitrary base field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-19T13:41:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stable homotopy groups",
      "framed motives",
      "motivic spectra",
      "hypercomplete etale",
      "realization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}