2004/07/31 by Osamu Iyama · 2 citations
Mathematics · #math.RT #math.AC #msc:16E30 #msc:16G70
published as Adv. Math. 210 (2007), no. 1, 22--50 · 25 pages. Final Version. To appear in Adv. Math
arxiv created 2006/05/21 · arxiv updated 2010/11/01
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.