{
  "id": "math/0407052",
  "version": "v3",
  "published": "2004-07-05T08:18:33.000Z",
  "updated": "2006-05-21T16:44:39.000Z",
  "title": "Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories",
  "authors": [
    "Osamu Iyama"
  ],
  "comment": "25 pages. Final Version. To appear in Adv. Math",
  "journal": "Adv. Math. 210 (2007), no. 1, 22--50",
  "categories": [
    "math.RT",
    "math.AC"
  ],
  "abstract": "Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on projective curves. In these Auslander-Reiten theories, the number `2' is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple functors of projective dimension `2'. For another, Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most beautiful situation in representation theory, which is closely related to McKay's observation on simple singularities. In this sense, usual Auslander-Reiten theory should be `2-dimensional' theory, and it be natural to find a setting for higher dimensional Auslander-Reiten theory from the viewpoint of representation theory and non-commutative algebraic geometry. We introduce maximal $(n-1)$-orthogonal subcategories as a natural domain of higher dimensional Auslander-Reiten theory which should be `$(n+1)$-dimensional'. We show that the $n$-Auslander-Reiten translation functor and the $n$-Auslander-Reiten duality can be defined quite naturally for such categories. Using them, we show that our categories have {\\it $n$-almost split sequences}, which give minimal projective resolutions of simple objects of projective dimension `$n+1$' in functor categories. We show that an invariant subring (of Krull-dimension `$n+1$') corresponding to a finite subgroup $G$ of ${\\rm GL}(n+1,k)$ has a natural maximal $(n-1)$-orthogonal subcategory. We give a classification of all maximal 1-orthogonal subcategories for representation-finite selfinjective algebras and representation-finite Gorenstein orders of classical type.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-05-21T16:44:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E30",
      "16G70"
    ],
    "keywords": [
      "higher dimensional auslander-reiten theory",
      "orthogonal subcategory",
      "maximal orthogonal subcategories",
      "representation theory",
      "minimal projective resolutions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7052I"
    }
  }
}