{
  "id": "1703.08725",
  "version": "v1",
  "published": "2017-03-25T18:13:53.000Z",
  "updated": "2017-03-25T18:13:53.000Z",
  "title": "Homological behavior of idempotent subalgebras and Ext algebras",
  "authors": [
    "Colin Ingalls",
    "Charles Paquette"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the algebra $\\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda algebra $Y(e):={\\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\\Gamma$. We also investigate the situation where both $A,\\Gamma$ have finite global dimension. When $A$ is Koszul finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-25T18:13:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E10",
      "16G10"
    ],
    "keywords": [
      "finite global dimension",
      "idempotent subalgebras",
      "ext algebras",
      "homological behavior",
      "long-standing cartan determinant conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}