{
  "id": "1909.07415",
  "version": "v1",
  "published": "2019-09-16T18:08:05.000Z",
  "updated": "2019-09-16T18:08:05.000Z",
  "title": "Chern Classes via Derived Determinant",
  "authors": [
    "Gleb Terentiuk"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Motivated by the Chern-Weil theory, we prove that for a given vector bundle $E$ on a smooth scheme $X$ over a field $k$ of any characteristic, the Chern classes of $E$ in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \\cite{i}, we present another proof by means of derived algebraic geometry. Also, for a scheme $X$ over a field $k$ of characteristic $p$ with a vector bundle $E$ we construct elements $c^{cris}_n (E, \\alpha(E)) \\in H_{dR}^{2n} (X) $ using an obstruction $\\alpha(E)$ to a lifting of $F^* E$ to a crystal modulo $p^2$ and prove that $c^{cris}_n (E, \\alpha(E)) = n! \\cdot c_{n}^{dR} (E)$, where $c_{n}^{dR} (E)$ are the Chern classes of $E$ in the de Rham cohomology and $F$ is the Frobenius map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-16T18:08:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chern classes",
      "derived determinant",
      "vector bundle",
      "smooth scheme",
      "characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}