{
  "id": "1405.5429",
  "version": "v2",
  "published": "2014-05-21T14:08:55.000Z",
  "updated": "2015-08-22T16:12:19.000Z",
  "title": "Homological dimensions for co-rank one idempotent subalgebras",
  "authors": [
    "Colin Ingalls",
    "Charles Paquette"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $\\Gamma = (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\\rm rad}A$, where ${\\rm rad}A$ is the Jacobson radical of $A$, or the graded Jacobson radical of $A$ if $A$ is positively graded. In this paper, we relate the homological dimensions of $A$ and $\\Gamma$, using the homological properties of $S_e$. First, if $S_e$ has no self-extensions of any degree, then the global dimension of $A$ is finite if and only if that of $\\Gamma$ is. On the other hand, if the global dimensions of both $A$ and $\\Gamma$ are finite, then $S_e$ cannot have self-extensions of degree greater than one, provided $A/{\\rm rad}A$ is finite dimensional.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-21T14:08:55.000Z",
      "abstract": "Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is not semiperfect but positively graded. We consider the idempotent subalgebra $\\Gamma = (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\\rm rad}A$, where ${\\rm rad}A$ is the Jacobson radical of $A$, or the graded Jacobson radical of $A$ if $A$ is not semiperfect but positively graded. In this paper, we relate the homological dimensions of $A$ and $\\Gamma$, using the homological properties of $S_e$. First, if $S_e$ has no self-extensions of any degree, then the global dimension of $A$ is finite if and only if that of $\\Gamma$ is. On the other hand, if the global dimensions of both $A$ and $\\Gamma$ are finite, it seems that $S_e$ cannot have self-extensions of degree greater than one. We prove this in the positively graded case, and for the semiperfect case, we get a partial answer: it holds when the projective dimension of $S_e$ is at most five.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-08-22T16:12:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E10",
      "16G10"
    ],
    "keywords": [
      "idempotent subalgebra",
      "homological dimensions",
      "global dimension",
      "jacobson radical"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.5429I"
    }
  }
}