Cluster tilting for higher Auslander algebras

Abstract

The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor ${\tau }_n$ plays an important role in the study of $n$-cluster tilting subcategories. We study the category $\MM_n$ of preinjective-like modules obtained by applying ${\tau }_n$ to injective modules repeatedly. We call a finite dimensional algebra $\Lambda$ \emph{$n$-complete} if $\MM_n=\add M$ for an $n$-cluster tilting object $M$. Our main result asserts that the endomorphism algebra $\End_Λ(M)$ is $(n+1)$-complete. This gives an inductive construction of $n$-complete algebras. For example, any representation-finite hereditary algebra ${\Lambda }^{(1)}$ is 1-complete. Hence the Auslander algebra ${\Lambda }^{(2)}$ of ${\Lambda }^{(1)}$ is 2-complete. Moreover, for any $n\ge 1$, we have an $n$-complete algebra ${\Lambda }^{(n)}$ which has an $n$-cluster tilting object $M^{(n)}$ such that $Λ^{(n+1)}=\End_{Λ^{(n)}}(M^{(n)})$. We give the presentation of ${\Lambda }^{(n)}$ by a quiver with relations. We apply our results to construct $n$-cluster tilting subcategories of derived categories of $n$-complete algebras.

42 pages. Typos are corrected

References in corpus (11)

• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Acyclic Calabi-Yau categories
• Cluster algebra structures and semicanonical bases for unipotent groups
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• Stable categories of higher preprojective algebras
• The Existence of Maximal $n$-Orthogonal Subcategories
• Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories
• Trivial Maximal 1-Orthogonal Subcategories For Auslander's 1-Gorenstein Algebras
• Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A
• n-representation-finite algebras and n-APR tilting
• Rigid objects in higher cluster categories

Cited by in corpus (5)

• Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories
• Trivial Maximal 1-Orthogonal Subcategories For Auslander's 1-Gorenstein Algebras
• 2-Auslander algebras associated with reduced words in Coxeter groups
• From Auslander Algebras to Tilted Algebras
• Transfer of stable equivalences of Morita type