{
  "id": "1904.08033",
  "version": "v1",
  "published": "2019-04-17T00:44:26.000Z",
  "updated": "2019-04-17T00:44:26.000Z",
  "title": "The module of vector-valued modular forms is Cohen-Macaulay",
  "authors": [
    "Richard Gottesman"
  ],
  "comment": "Six pages",
  "categories": [
    "math.NT",
    "math.AC"
  ],
  "abstract": "Let $H$ denote a finite index subgroup of the modular group $\\Gamma$ and let $\\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\\rho)$ denote the collection of holomorphic vector-valued modular forms for $\\rho$ and let $M(H)$ denote the collection of modular forms on $H$. Then $M(\\rho)$ is a $\\textbf{Z}$-graded $M(H)$-module. It has been proven that $M(\\rho)$ may not be projective as a $M(H)$-module. We prove that $M(\\rho)$ is Cohen-Macaulay as a $M(H)$-module. We also explain how to apply this result to prove that if $M(H)$ is a polynomial ring then $M(\\rho)$ is a free $M(H)$-module of rank $\\textrm{dim } \\rho.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-17T00:44:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F03",
      "13C14",
      "13H10"
    ],
    "keywords": [
      "cohen-macaulay",
      "finite index subgroup",
      "holomorphic vector-valued modular forms",
      "finite-dimensional complex representation",
      "modular group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}