Abstract

Abstract. We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher Auslander-Reiten theory on them. Auslander-Reiten theory, especially the concept of almost split sequences and their existence theorem, is fundamental to study categories which appear in representation theory, for example, modules over artin algebras [ARS][GR][Ri], their functorially finite subcategories [AS][S], their derived categories [H], Cohen-Macaulay modules over Cohen-Macaulay rings [Y], lattices over orders [A2,3][RS], and coherent sheaves on projective curves [AR2][GL]. In these Auslander-Reiten theory, the number ‘2 ’ is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple objects of projective dimension ‘2 ’ in functor categories. For another, Cohen-Macaulay rings and orders of Krull-dimension ‘2 ’ have fundamental sequences and provide us one of the most beautiful situation in representation theory [A4][E][RV][Y], which is closely related to McKay’s observation on simple singularities [Ma]. In this sense, usual Auslander-Reiten theory should be ‘2-dimensional ’ theory, and it would have natural importance to search a domain of higher Auslander-Reiten theory from the viewpoint of representation

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