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

Auslander--Reiten theory in extriangulated categories

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander--Reiten theory for extriangulated categories. This unifies Auslander--Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost-split extensions becomes equivalent to that of an Auslander--Reiten--Serre duality. We also show that existence of almost-split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category $\underline{\mathscr{C}}$ of an extriangulated category $\mathscr{C}$ is a $\tau$-category if $\mathscr{C}$ has almost split extensions and source morphisms. This gives various consequences on $\underline{\mathscr{C}}$, including Igusa--Todorov's Radical Layers Theorem. In particular, the associated completely graded category of $\underline{\mathscr{C}}$ is equivalent to the complete mesh category of the Auslander-Reiten species of $\underline{\mathscr{C}}$.Comment: Revised, 40 pages. Section 6 and 7 adde

Covering techniques in Auslander-Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.Comment: Minor modifications. Final version to appear in the Journal of Pure and Applied Algebr

Gabriel-Roiter inclusions and Auslander-Reiten theory

Let $\Lambda$ be an artin algebra. The aim of this paper is to outline a strong relationship between the Gabriel-Roiter inclusions and the Auslander-Reiten theory. If $X$ is a Gabriel-Roiter submodule of $Y,$ then $Y$ is shown to be a factor module of an indecomposable module $M$ such that there exists an irreducible monomorphism $X \to M$. We also will prove that the monomorphisms in a homogeneous tube are Gabriel-Roiter inclusions, provided the the tube contains a module whose endomorphism ring is a division ring

Auslander-Reiten theory of small half quantum groups

For the small half quantum groups and we show that the components of the stable Auslander-Reiten quiver containing gradable modules are of the form Z[A_\infty]Comment: 10 page

Auslander-Reiten theory for simply connected differential graded algebras

Peter Jorgensen introduced the Auslander-Reiten quiver of a simply connected Poincare duality space. He showed that its components are of the form ZA_infty and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d-1 such components. In this thesis we show that this is the only case where finitely many components appear. More precisely, we construct families of modules, where for each family, each module lies in a different component. Depending on the cohomology dimensions of the differential graded algebras which appear, this is either a discrete family or an n-parameter family for all n.Comment: 58 pages, doctoral thesis University of Paderborn (2007