Abstract:
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.

Abstract:
We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function $r_\Lambda:\mod\Lambda\to\nnn_{\ge0}$ which is a natural extension of Auslander's representation dimension.

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 $\End_\Lambda(M)$ 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.

Abstract:
We will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $\tau$-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.

Abstract:
We will show that there exists a close relationship between quasi-hereditary algebras of Cline-Parshall-Scott and the rejection theory from the viewpoint of the approximation theory of Auslander-Smalo. As an application, we will solve two open problems. One concerns the representation dimension of artin algebras introduced by M. Auslander about 30 years ago, and another concerns the Solomon zeta functions of orders introduced by L. Solomon about 25 years ago. It will turn out that the rejection theory relates these two quite different problems with each other closely.

Abstract:
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.

Abstract:
We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.

Abstract:
We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of non-commutative crepant resolutions.

Abstract:
We introduce (n+1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+1)-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)-preprojective algebra is (n+1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)-cluster tilting object. We show that even if the (n+1)-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n \in {1,2}) the results above still hold for the stable category of Cohen-Macaulay modules.

Abstract:
In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with `silting mutation' by generalizing the theory of Riedtmann-Schofield and Happel-Unger. We show that iterated silting mutation act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally we give a bijection between silting subcategories and certain t-structures.