{
  "id": "1710.06129",
  "version": "v1",
  "published": "2017-10-17T07:12:09.000Z",
  "updated": "2017-10-17T07:12:09.000Z",
  "title": "Higher Nerves of Simplicial Complexes",
  "authors": [
    "Hailong Dao",
    "Joseph Doolittle",
    "Ken Duna",
    "Bennet Goeckner",
    "Brent Holmes",
    "Justin Lyle"
  ],
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "We investigate generalized notions of the nerve complex for a collection of subsets of a topological space. For a simplicial complex $\\Delta$, we show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[\\Delta]$ as well as the $f$-vector and $h$-vector of $\\Delta$. We develop a strengthened version of the higher nerve complexes, called the nervous system of $\\Delta$, that allows one to reconstruct $\\Delta$. We present, as an application, a formula for computing regularity of monomial ideals and a new definition of \"combinatorial depth\" for local schemes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-17T07:12:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E40",
      "05E45",
      "13C15",
      "13D03"
    ],
    "keywords": [
      "simplicial complex",
      "higher nerve complexes determine",
      "monomial ideals",
      "combinatorial depth",
      "local schemes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}