Mathematics – Representation Theory
Scientific paper
2008-09-29
Adv. Math. 226 (2011), no. 1, 1--61
Mathematics
Representation Theory
42 pages. Typos are corrected
Scientific paper
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 $\End_\Lambda(M)$ 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 $n\ge1$, we have an $n$-complete algebra $\Lambda^{(n)}$ which has an $n$-cluster tilting object $M^{(n)}$ such that $\Lambda^{(n+1)}=\End_{\Lambda^{(n)}}(M^{(n)})$. 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.
No associations
LandOfFree
Cluster tilting for higher Auslander algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Cluster tilting for higher Auslander algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cluster tilting for higher Auslander algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-48799