Trivial maximal 1-orthogonal subcategories for auslander 1-gorenstein algebras

摘要

Let Λ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If Λ admits a trivial maximal 1-orthogonal subcategory of Λ, then, for any indecomposable module M Λ, the projective dimension of M is equal to one if and only if its injective dimension is also equal to one, and M is injective if the projective dimension of M is equal to two. In this case, we further get that Λ is a tilted algebra.