{
  "id": "2408.15893",
  "version": "v1",
  "published": "2024-08-28T16:04:09.000Z",
  "updated": "2024-08-28T16:04:09.000Z",
  "title": "Azumaya algebras over unramifed extensions of function fields",
  "authors": [
    "Mohammed Moutand"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a smooth variety over a field $K$ with function field $K(X)$. Using the interpretation of the torsion part of the \\'etale cohomology group $H_{\\text{\\'et}}^2(K(X), \\mathbb{G}_m)$ in terms of Milnor-Quillen algebraic $K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over $X$, there exist cohomological Brauer classes in $\\operatorname{Br}'(X)$ that are representable by Azumaya algebras on $X$. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-28T16:04:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function field",
      "azumaya algebras",
      "unramifed extensions",
      "etale cohomology group",
      "milnor-quillen algebraic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}