{
  "id": "1907.12620",
  "version": "v1",
  "published": "2019-07-29T20:03:45.000Z",
  "updated": "2019-07-29T20:03:45.000Z",
  "title": "Algebraic $h$-vectors of simplicial complexes through local cohomology, part 1",
  "authors": [
    "Connor Sawaske"
  ],
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Given an infinite field $\\mathbb{k}$ and a simplicial complex $\\Delta$, a common theme in studying the $f$- and $h$-vectors of $\\Delta$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $\\mathbb{k}[\\Delta]$ modulo a generic linear system of parameters $\\Theta$. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of $h_{d-1}^\\mathfrak{a}(\\Delta)$, the dimension over $\\mathbb{k}$ in degree $d-1$ of $\\mathbb{k}[\\Delta]/(\\Theta)$, for any complex $\\Delta$ of dimension $d-1$. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-29T20:03:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F55",
      "05E45"
    ],
    "keywords": [
      "simplicial complex",
      "local cohomology",
      "hilbert series",
      "generic linear system",
      "common theme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}