{
  "id": "2101.09803",
  "version": "v1",
  "published": "2021-01-24T21:16:38.000Z",
  "updated": "2021-01-24T21:16:38.000Z",
  "title": "The structure of Koszul algebras defined by four quadrics",
  "authors": [
    "Paolo Mantero",
    "Matthew Mastroeni"
  ],
  "categories": [
    "math.AC"
  ],
  "abstract": "Avramov, Conca, and Iyengar ask whether $\\beta_i^S(R) \\leq \\binom{g}{i}$ for all $i$ when $R=S/I$ is a Koszul algebra minimally defined by $g$ quadrics. In recent work, we give an affirmative answer to this question when $g \\leq 4$ by completely classifying the possible Betti tables of Koszul algebras defined by height-two ideals of four quadrics. Continuing this work, the current paper proves a structure theorem for Koszul algebras defined by four quadrics. We show that all these Koszul algebras are LG-quadratic, proving that an example of Conca of a Koszul algebra that is not LG-quadratic is minimal in terms of number of defining equations. We then characterize precisely when these rings are absolutely Koszul, and establish the equivalence of the absolutely Koszul and Backelin--Roos properties up to field extensions for such rings (in characteristic zero). The combination of the above paper with the current one provides a fairly complete picture of all Koszul algebras defined by $g \\leq 4$ quadrics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-24T21:16:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "koszul algebra",
      "absolutely koszul",
      "iyengar ask",
      "characteristic zero",
      "lg-quadratic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}