The Existence of Maximal $n$-Orthogonal Subcategories

TL;DR

This paper investigates conditions for the existence of maximal n-orthogonal subcategories in various algebraic contexts, linking properties of modules, Gorenstein conjectures, and tensor products.

Contribution

It provides necessary conditions, characterizations, and connections between maximal n-orthogonal subcategories and algebraic properties like Gorenstein symmetry and module complexity.

Findings

Necessary conditions for maximal (n-1)-orthogonal subcategories in (n-1)-Auslander algebras.

Characterization of maximal 1-orthogonal subcategories in almost hereditary algebras.

Relation between module periodicity and maximal n-orthogonal subcategories in Gorenstein algebras.

Abstract

For an $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n$, we give some necessary conditions for $\Lambda$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $\Lambda$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $\Lambda$ with global dimension 2, we prove that $\Lambda$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $\Lambda$-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 $^{\bot}T$ for a cotilting module $T$. For a Gorenstein algebra, we prove that all non-projective direct summands of a maximal $n$-orthogonal module are $\Omega ^n\tau$-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.