The Existence of Maximal $n$-Orthogonal Subcategories

Details The Existence of Maximal $n$-Orthogonal Subcategories The Existence of Maximal $n$-Orthogonal Subcategories

2009-03-04

arxiv.org/abs/0903.0758v1

Mathematics

Representation Theory

19 pages, to appear in Journal of Algebra

Scientific paper

10.1016/j.jalgebra.2009.01.036

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.

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