Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories

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 CohenMacaulay 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 AuslanderReiten 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 theory and non-commutative algebraic geometry (e.g. [V][ArS][GL]). In this paper, we introduce (n− 1)-orthogonal subcategories as a natural domain of higher Auslander-Reiten theory (§2.2) which should be ‘(n + 1)-dimensional’. We show that higher Auslander-Reiten translation and higher Auslander-Reiten duality can be defined quite naturally for such categories (§2.3,§2.3.1). Using them, we show that our categories have n-almost split sequences (§3.1,§3.3.1), which are completely new generalization of usual almost split sequences and give minimal projective resolutions of simple objects of projective dimension ‘n+1’ in functor categories. We also show the existence of higher dimensional analogy of fundamental sequences for Cohen-Macaulay rings and orders of Krull-dimension ‘n+ 1’ (§3.4.3). We show that an invariant subring (of Krulldimension ‘n + 1’) corresponding to a finite subgroup G of GL(n + 1, k) has a natural maximal (n − 1)-orthogonal subcategory (§2.5). In the final section, we give a classification of all maximal 1-orthogonal subcategories for representation-finite selfinjective algebras and representation-finite Gorenstein orders of classical type (§4.2.2). We show that the number of such subcategories is related to Catalan number (§4.2.3). These results would suggest us that our higher Auslander-Reiten theory is far from abstract non-sense. 2000 Mathematics Subject Classification. Primary 16E30; Secondary 16G70