Higher Auslander algebras admitting trivial maximal orthogonal subcategories (original) (raw)

For an Artinian (n − 1)-Auslander algebra Λ with global dimension n(≥ 2), we show that if Λ admits a trivial maximal (n − 1)-orthogonal subcategory of mod Λ, then Λ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a result, for an Artinian Auslander algebra with global dimension 2, if Λ admits a trivial maximal 1-orthogonal subcategory of mod Λ, then Λ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra Λ over an algebraically closed field K, we show that Λ is a basic and connected (n − 1)-Auslander algebra Λ with global dimension n(≥ 2) admitting a trivial maximal (n − 1)-orthogonal subcategory of mod Λ if and only if Λ is given by the quiver: