Cluster tilting for higher Auslander algebras

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.Comment: 42 pages. Typos are correcte

Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories

Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on projective curves. In these Auslander-Reiten theories, the number `2' is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple functors of projective dimension `2'. For another, Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most beautiful situation in representation theory, which is closely related to McKay's observation on simple singularities. In this sense, usual Auslander-Reiten theory should be `2-dimensional' theory, and it be natural to find a setting for higher dimensional Auslander-Reiten theory from the viewpoint of representation theory and non-commutative algebraic geometry. We introduce maximal $(n-1)$-orthogonal subcategories as a natural domain of higher dimensional Auslander-Reiten theory which should be `$(n+1)$-dimensional'. We show that the $n$-Auslander-Reiten translation functor and the $n$-Auslander-Reiten duality can be defined quite naturally for such categories. Using them, we show that our categories have {\it $n$-almost split sequences}, which give minimal projective resolutions of simple objects of projective dimension `$n+1$' in functor categories. We show that an invariant subring (of Krull-dimension `$n+1$') corresponding to a finite subgroup $G$ of ${\rm GL}(n+1,k)$ has a natural maximal $(n-1)$-orthogonal subcategory. We give a classification of all maximal 1-orthogonal subcategories for representation-finite selfinjective algebras and representation-finite Gorenstein orders of classical type.Comment: 25 pages. Final Version. To appear in Adv. Mat

Tilting Cohen-Macaulay representations

This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster) categories of finite dimensional algebras.Comment: To appear in the ICM 2018 proceeding

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. We give homological characterizations of Auslander algebras, especially an answer to a question of M. Artin. They are also closely related to Auslander's representation dimension of artin algebras and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities.Comment: 29 pages. Final Version. To appear in Adv. Mat