{
  "id": "1510.05124",
  "version": "v1",
  "published": "2015-10-17T12:44:25.000Z",
  "updated": "2015-10-17T12:44:25.000Z",
  "title": "Monic monomial representations I Gorenstein-projective modules",
  "authors": [
    "Xiu-Hua Luo",
    "Pu Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "For a finite-dimensional algebra $A$ over field $k$, a finite acyclic quiver $Q$, and an ideal $I$ of the path algebra $kQ$ generated by monomial relations, let $\\Lambda: = A\\otimes_k kQ/I$. We introduce the monic representations of $(Q, I)$ over $A$. By the equivalence $\\m\\mbox{-}{\\rm mod}\\cong {\\rm rep}(Q, I, A)$, we also have the notion of monic $\\m$-module. We give properties of the structural maps of monic representations, and prove that the category of the monic representations of $(Q, I)$ over $A$ is a resolving subcategory of ${\\rm rep}(Q, I, A)$. We introduce the condition ${\\rm(G)}$. The main result of this paper claims that a $\\m$-module is Gorenstein-projective if and only if it is a monic module satisfying ${\\rm(G)}$. This provides an inductive construction of Gorenstein-projective modules. As consequences, the monic $\\m$-modules are exactly the projective $\\m$-modules if and only if $A$ is semisimple, and the monic $\\m$-modules are exactly the Gorenstein-projective $\\m$-modules if and only if $A$ is selfinjective.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-17T12:44:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16E65",
      "16G50",
      "16G60"
    ],
    "keywords": [
      "monic monomial representations",
      "gorenstein-projective modules",
      "monic representations",
      "finite acyclic quiver",
      "monic module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151005124L"
    }
  }
}