{
  "id": "1709.09097",
  "version": "v1",
  "published": "2017-09-26T15:50:35.000Z",
  "updated": "2017-09-26T15:50:35.000Z",
  "title": "Comparing motives of smooth algebraic varieties",
  "authors": [
    "Grigory Garkusha"
  ],
  "categories": [
    "math.AG",
    "math.CT",
    "math.KT"
  ],
  "abstract": "Given a perfect field of exponential characteristic $e$ and a functor $f:\\mathcal A\\to\\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\\mathbb Z_{\\mathcal A}(q)[1/e]\\to\\mathbb Z_{\\mathcal B}(q)[1/e],\\quad q\\geq 0,$$ are quasi-isomorphisms locally in the Nisnevich topology, it is proved that for every $k$-smooth algebraic variety $X$ the morphisms of twisted motives of $X$ with $\\mathbb Z[1/e]$-coefficients $$M_{\\mathcal A}(X)(q)\\otimes\\mathbb Z[1/e]\\to M_{\\mathcal B}(X)(q)\\otimes\\mathbb Z[1/e]$$ are quasi-isomorphisms locally in the Nisnevich topology. Furthermore, it is shown that the induced functors between triangulated categories of motives $$DM_{\\mathcal A}^{eff}(k)[1/e]\\to DM_{\\mathcal B}^{eff}(k)[1/e],\\quad DM_{\\mathcal A}(k)[1/e]\\to DM_{\\mathcal B}(k)[1/e]$$ are equivalences. As an application, the Cor-, $K_0^\\oplus$-, $K_0$- and $\\mathbb K_0$-motives of smooth algebraic varieties with $\\mathbb Z[1/e]$-coefficients are locally quasi-isomorphic to each other. Moreover, their triangulated categories of motives with $\\mathbb Z[1/e]$-coefficients are shown to be equivalent. Another application is given for the bivariant motivic spectral sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-26T15:50:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth algebraic variety",
      "nisnevich topology",
      "bivariant motivic spectral sequence",
      "coefficients",
      "symmetric monoidal strict"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}