{
  "id": "1907.12710",
  "version": "v1",
  "published": "2019-07-30T02:32:06.000Z",
  "updated": "2019-07-30T02:32:06.000Z",
  "title": "Depth of an initial ideal",
  "authors": [
    "Takayuki Hibi",
    "Akiyoshi Tsuchiya"
  ],
  "comment": "4 pages",
  "categories": [
    "math.AC"
  ],
  "abstract": "Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \\ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of the integers $0 \\leq r \\leq d$, there exists a monomial order $<_r$ on $S$ with ${\\rm depth} (S/{\\rm in}_{<_r}(I)) = r$, where ${\\rm in}_{<_r}(I)$ is the initial ideal of $I$ with respect to $<_r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T02:32:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13P10"
    ],
    "keywords": [
      "initial ideal",
      "arbitrary integer",
      "monomial order",
      "homogeneous ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}