{
  "id": "2403.05232",
  "version": "v1",
  "published": "2024-03-08T11:45:33.000Z",
  "updated": "2024-03-08T11:45:33.000Z",
  "title": "Chains of model structures arising from modules of finite Gorenstein dimension",
  "authors": [
    "Nan Gao",
    "Xue-Song Lu",
    "Pu Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $n$ be a non-negative integer. For any ring $R$, the pair \\ $(\\mathcal {PGF}_n, \\ \\mathcal P_n^\\perp \\cap \\mathcal {PGF}^{\\perp})$ proves to be a complete and hereditary cotorsion pair in $R$-Mod, where $\\mathcal {PGF}$ is the class of PGF modules, introduced by J. \\v{S}aroch and J. \\v{S}\\'{t}ov\\'{i}\\v{c}ek, and \\ $\\mathcal {PGF}_n$ is the class of $R$-modules of PGF dimension $\\le n$. For Artin algebra $R$, it is proved that \\ $(\\mathcal {GP}_n, \\ \\mathcal P_n^\\perp \\cap \\mathcal P^{<\\infty})$ is a complete and hereditary cotorsion pair in $R$-Mod, where $\\mathcal {GP}_n$ is the class of modules of Gorenstein projective dimension $\\le n$, and $\\mathcal P^{<\\infty}$ is the class of modules of finite projective dimension. The two chains of cotorsion pairs induce two chains of hereditary Hovey triples \\ $(\\mathcal {PGF}_n, \\ \\mathcal P_n^\\perp, \\ \\mathcal {PGF}^{\\perp})$ and \\ $(\\mathcal {GP}_n, \\ \\mathcal P_n^\\perp, \\ \\mathcal P^{<\\infty})$, and the corresponding abelian model structures on $R$-Mod in the same chain have the same homotopy category, up to triangle equivalence. The corresponding results in exact categories $\\mathcal {PGF}_n$, \\ $\\mathcal {GP}_n$, \\ $\\mathcal {GF}_n$ and in $\\mathcal {PGF}^{<\\infty}$, $\\mathcal {GP}^{<\\infty}$ and $\\mathcal {GF}^{<\\infty}$, are also obtained. As a byproduct, $\\mathcal{PGF} = \\mathcal {GP}$ for a ring $R$ if and only if $\\mathcal{PGF}^\\perp\\cap\\mathcal{GP}_n=\\mathcal P_n$ for some non-negative integer $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-08T11:45:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E30",
      "18N40",
      "16E10",
      "16E65",
      "16G50"
    ],
    "keywords": [
      "finite gorenstein dimension",
      "model structures arising",
      "hereditary cotorsion pair",
      "cotorsion pairs induce",
      "corresponding abelian model structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}