Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories

Abstract

For an Artinian $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n(\ge 2)$, we show that if $\Lambda$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\mathrm{mod}\Lambda$, then $\Lambda$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\mathrm{mod}\Lambda$ is at most $n-1$. As a result, for an Artinian Auslander algebra with global dimension 2, if $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\mathrm{mod}\Lambda$, then $\Lambda$ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra $\Lambda$ over an algebraically closed field $K$, we show that $\Lambda$ is a basic and connected $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n(\ge 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\mathrm{mod}\Lambda$ if and only if $\Lambda$ is given by the quiver: $$\xymatrix{1 & \ar[l]_{β_{1}} 2 & \ar[l]_{β_{2}} 3 & \ar[l]_{β_{3}} ... & \ar[l]_{β_{n}} n+1} $$ modulo the ideal generated by $\{{\beta }_i{\beta }_{i+1}|1\le i\le n-1\}$. As a consequence, we get that a finite-dimensional algebra over an algebraically closed field $K$ is an $(n-1)$-Auslander algebra with global dimension $n(\ge 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory if and only if it is a finite direct product of $K$ and $\Lambda$ as above. Moreover, we give some necessary condition for an Artinian Auslander algebra admitting a non-trivial maximal 1-orthogonal subcategory.

25 pages. This version is a combination of the orginal version of this paper with "From Auslander Algebras to Tilted Algebras" (arXiv:0903.0760). The latter paper has been withdrawn

References in corpus (2)

• The Existence of Maximal $n$-Orthogonal Subcategories
• From Auslander Algebras to Tilted Algebras

Cited by in corpus (1)

• Trivial Maximal 1-Orthogonal Subcategories For Auslander's 1-Gorenstein Algebras