{
  "id": "1801.06388",
  "version": "v1",
  "published": "2018-01-19T13:01:37.000Z",
  "updated": "2018-01-19T13:01:37.000Z",
  "title": "A note on multivariable $(\\varphi,Γ)$-modules",
  "authors": [
    "Elmar Große-Klönne"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F/{\\mathbb Q}_p$ be a finite field extension, let $k$ be a field of characteristic $p$. Fix a Lubin Tate group $\\Phi$ for $F$ and let $\\Gamma\\times\\cdots\\times\\Gamma$ with $\\Gamma={\\mathcal O}_F^{\\times}$ act on $k[[t_1,\\ldots,t_n]][\\prod_it_i^{-1}]$ by letting $\\gamma_i$ (in the $i$-th factor $\\Gamma$) act on $t_i$ by insertion of $t_i$ into the power series attached to $\\gamma_i$ by $\\Phi$. We show that $k[[t_1,\\ldots,t_n]][\\prod_it_i^{-1}]$ admits no non-trivial ideal stable under $\\Gamma$, thereby generalizing a result of Z\\'{a}br\\'{a}di (who had treated the case where $\\Phi$ is the multiplicative group). We then discuss applications to $(\\varphi,\\Gamma)$-modules over $k[[t_1,\\ldots,t_n]][\\prod_it_i^{-1}]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-19T13:01:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite field extension",
      "lubin tate group",
      "power series",
      "th factor",
      "multivariable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}