{
  "id": "1808.02709",
  "version": "v1",
  "published": "2018-08-08T10:21:12.000Z",
  "updated": "2018-08-08T10:21:12.000Z",
  "title": "$d$-Auslander-Reiten sequences in subcategories",
  "authors": [
    "Francesca Fedele"
  ],
  "comment": "25 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\Phi$ be a finite dimensional algebra over an algebraically closed field. Kleiner described the Auslander-Reiten sequences in a precovering extension closed subcategory $\\mathcal{X}\\subseteq\\text{mod }\\Phi$. If $X\\in\\mathcal{X}$ is an indecomposable such that $\\text{Ext}^1(X,\\mathcal{X})\\neq 0$ and $\\sigma X$ is the unique indecomposable direct summand of the $\\mathcal{X}$-cover $g:Y\\rightarrow D\\text{Tr } X$ such that $\\text{Ext}^1(X,\\sigma X)\\neq 0$, then there is an Auslander-Reiten sequence in $\\mathcal{X}$ of the form \\begin{align*} \\epsilon: 0\\rightarrow \\sigma X\\rightarrow X'\\rightarrow X\\rightarrow 0. \\end{align*} Moreover, when $\\text{End } (X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form \\begin{align*} \\delta: 0\\rightarrow Y\\rightarrow Y'\\xrightarrow{\\eta} X\\rightarrow 0 \\end{align*} is such that $\\eta$ is right almost split in $\\mathcal{X}$, and the pushout of $\\delta$ along $g$ gives an Auslander-Reiten sequence in $\\text{mod }\\Phi$ ending at $X$. In this paper, we give higher dimensional generalisations of this. Let $d\\geq 1$ be an integer. A $d$-cluster tilting subcategory $\\mathcal{F}\\subseteq\\text{mod }\\Phi$ plays the role of a higher $\\text{mod }\\Phi$. Such an $\\mathcal{F}$ is a $d$-abelian category, where kernels and cokernels are replaced by complexes of $d$ objects and short exact sequences by complexes of $d+2$ objects. We give higher versions of the above results for an additive ''$d$-extension closed'' subcategory $\\mathcal{X}$ of $\\mathcal{F}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-08T10:21:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "auslander-reiten sequence",
      "non-split short exact sequence",
      "finite dimensional algebra",
      "unique indecomposable direct summand",
      "higher dimensional generalisations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}