Auslander correspondence

Abstract

We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander–Reiten theory [O. Iyama, Higher dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22–50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195–222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770].