{
  "id": "2008.04178",
  "version": "v1",
  "published": "2020-08-10T15:01:24.000Z",
  "updated": "2020-08-10T15:01:24.000Z",
  "title": "On the Monomorphism Category of $n$-Cluster Tilting Subcategories",
  "authors": [
    "Javad Asadollahi",
    "Rasool Hafezi",
    "Somayeh Sadeghi"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\\rm mod}\\mbox{-}\\Lambda$, where $\\Lambda$ is an artin algebra. Let $\\mathcal{S}(\\mathcal{M})$ denotes the full subcategory of $\\mathcal{S}(\\Lambda)$, the submodule category of $\\Lambda$, consisting of all monomorphisms in $\\mathcal{M}$. We construct two functors from $\\mathcal{S}(\\mathcal{M})$ to ${\\rm mod}\\mbox{-}\\underline{\\mathcal{M}}$, the category of finitely presented (coherent) additive contravariant functors on the stable category of $\\mathcal{M}$. We show that these functors are full, dense and objective. So they induce equivalences from the quotient categories of the submodule category of $\\mathcal{M}$ modulo their respective kernels. Moreover, they are related by a syzygy functor on the stable category of ${\\rm mod}\\mbox{-}\\underline{\\mathcal{M}}$. These functors can be considered as a higher version of the two functors studied by Ringel and Zhang [RZ] in the case $\\Lambda=k[x]/{\\langle x^n \\rangle}$ and generalized later by Eir\\'{i}ksson [E] to self-injective artin algebras. Several applications will be provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-10T15:01:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E99",
      "18E10",
      "18G25",
      "16D90"
    ],
    "keywords": [
      "cluster tilting subcategory",
      "monomorphism category",
      "submodule category",
      "stable category",
      "quotient categories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}