Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories

Abstract

For an Artinian $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n(\geq 2)$, we show that if $\Lambda$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\mod\Lambda$, then $\Lambda$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\mod\Lambda$ 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 $\mod\Lambda$, 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 $n(\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\mod\Lambda$ if and only if $\Lambda$ is given by the quiver: $$\xymatrix{1 & \ar[l]_{\beta_{1}} 2 & \ar[l]_{\beta_{2}} 3 & \ar[l]_{\beta_{3}} ... & \ar[l]_{\beta_{n}} n+1} $$ modulo the ideal generated by $\{\beta_{i}\beta_{i+1}| 1\leq i\leq n-1 \}$. 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 $n(\geq 2)$ 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.