The existence of maximal n-orthogonal subcategories

For an (n−1)-Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal (n−1)-orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n−1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable Λ-module M, M is injective is equivalent to that the reduced grade of M is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence of maximal n-orthogonal subcategories of for a cotilting module T. For a Gorenstein algebra, we prove that all non-projective direct summands of a maximal n-orthogonal module are Ωnτ-periodic. In addition, we study the relation between the complexity of modules and the existence of maximal n-orthogonal subcategories for the tensor product of two finite-dimensional algebras.