{
  "id": "math/9912066",
  "version": "v1",
  "published": "1999-12-08T20:46:01.000Z",
  "updated": "1999-12-08T20:46:01.000Z",
  "title": "Irreducible components of characteristic varieties",
  "authors": [
    "Gregory G. Smith"
  ],
  "comment": "20 pages",
  "journal": "Journal of Pure and Applied Algebra 165 (2001) 291-306",
  "doi": "10.1016/S0022-4049(01)00109-8",
  "categories": [
    "math.AG",
    "math.AP",
    "math.RA"
  ],
  "abstract": "We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial differential equations defined from a suitable filtration of the Weyl algebra $A_{n}(k)$. This generalizes an important consequence of the fact that a characteristic variety defined from the order filtration is involutive. More explicitly, we consider a filtration of $A_{n}(k)$ induced by any vector $(u,v) \\in {\\mathbb Z}^{n}\\times {\\mathbb Z}^{n}$ such that the associated graded algebra is the commutative polynomial ring in $2n$ indeterminates. The order filtration is the special case $(u,v) = (0,1)$. Any finitely generated left $A_{n}(k)$-module $M$ has a good filtration with respect to $(u,v)$ and this gives rise to a characteristic variety $\\Ch_{(u,v)}(M)$ which depends only on $(u,v)$ and $M$. When $(u,v) = (0,1)$, the characteristic variety is involutive and this implies that its irreducible components have dimension at least $n$. In general, the characteristic variety may fail to be involutive, but we are still able to prove that each irreducible component of $\\Ch_{(u,v)}(M)$ has dimension at least $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-12-08T20:46:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "characteristic variety",
      "irreducible component",
      "order filtration",
      "linear partial differential equations",
      "weyl algebra"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....12066S"
    }
  }
}