The Dual Quiver of the Auslander–Reiten Quiver of Path Algebras

Let Q be a finite quiver of type A _n , n ≥ 1, D _n , n ≥ 4, E ₆, E ₇ and E ₈, σ ∈ Aut(Q), k be an algebraic closed field whose characteristic does not divide the order of σ. In this article, we prove that the dual quiver [(G_Q)\tilde]\widetilde{\Gamma_{Q}} of the Auslander–Reiten quiver Γ _Q of kQ, the Auslander–Reiten quiver of kQ#kás?kQ\#k\langle\sigma\rangle, and the Auslander–Reiten quiver G_[(Q)\tilde]\Gamma_{\widetilde{Q}} of k[(Q)\tilde]k\widetilde{Q}, where [(Q)\tilde]\widetilde{Q} is the dual quiver of Q, are isomorphic.     

τ-Categories II: Nakayama Pairs and Rejective Subcategories

We study Nakayama pairs in τ-categories, which are pairs of objects connected by a certain diagram of Auslander–Reiten sequences. Using them, we naturally introduce orderlikeτ-categories. Then we study rejective subcategories of τ-categories. A typical example of orderlike τ-categories is given by the category of lattices over an order. In this case, its rejective subcategories correspond bijectively to its overrings. Presented by K. Roggenkamp Mathematics Subject Classifications (2000) Primary: 16G30; secondary: 16E65, 16G70, 18E05. Osamu Iyama: Current address: Department of Mathematics, University of Hyogo, Himeji 671-2201, Japan. e-mail: iyama@sci.u-hyogo.ac.jp     

The Auslander–Reiten Quiver of a Poincaré Duality Space

In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincaré duality space, each component of the Auslander–Reiten quiver is isomorphic to . Presented by Yuri Drozd     

Hom-configurations and noncrossing partitions

We study maximal Hom-free sets in the τ[2]-orbit category C(Q) of the bounded derived category for the path algebra associated to a Dynkin quiver Q, where τ denotes the Auslander–Reiten translation and [2] denotes the square of the shift functor. We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom _≤0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutation of maximal Hom-free sets in C(Q)\mathcal {C}(Q) and prove that the graph of these mutations is connected.     

Locally finite triangulated categories

A k-linear triangulated category is called locally finite provided for any indecomposable object Y in . It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form :

Full-size image (<1K)

By using this, we prove that the Auslander–Reiten quiver of any locally finite triangulated category is of the form , where Δ is a Dynkin diagram and G is an automorphism group of . For most automorphism groups G, the triangulated categories with as their Auslander–Reiten quivers are constructed. In particular, a triangulated category with as its Auslander–Reiten quiver is constructed.     

Lattice-finite Rings

We study a one-dimensional analogue of representation-finite rings. For a left Noetherian semilocal ring R, we define an R-lattice to be a finitely generated R-module with zero socle. We call R lattice-finite if the number of isomorphism classes of indecomposable R-lattices is finite. Under this assumption, we give several equivalent criteria for the existence of Auslander–Reiten sequences in the category of R-lattices. A necessary condition is that the maximal left quotient ring of R is semisimple, and the main sufficient criterion states that R admits a semiperfect semiprime Asano left overorder. Presented by I. Reiten Mathematics Subject Classifications (2000) Primary: 16G70, 16G30; secondary: 16G60.     

Auslander–Reiten Quivers and the Coxeter Complex

Let Q be a quiver of type ADE. We construct the corresponding Auslander–Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. In A_n case, we recover special wiring diagrams. Presented by R. RentschlerMathematics Subject Classifications (2000) 16G70, 17B10, 20F55.     

Decompositions of degenerations

We develop a method to refine a given degeneration of modules. This enables us to give a new proof of the equivalence of the partial orders ≤_deg and ≤_ext in the case that the algebra Λ has a directed Auslander–Reiten quiver. Received: 11 November 1999     

Families of Auslander–Reiten components for simply connected differential graded algebras

Peter Jørgensen introduced the Auslander–Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form ${{\mathbb {Z}}A_\infty}Peter J?rgensen introduced the Auslander–Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form \mathbb ZA_￥{{\mathbb {Z}}A_\infty} and that the Auslander–Reiten quiver of a d-dimensional sphere consists of d − 1 such components. We show that this is essentially 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.     

Auslander–Reiten sequences on schemes

Auslander–Reiten sequences are the central item of Auslander–Reiten theory, which is one of the most important techniques for the investigation of the structure of abelian categories. This note considers X, a smooth projective scheme of dimension at least 1 over the field k, and , an indecomposable coherent sheaf on X. It is proved that in the category of quasi-coherent sheaves on X, there is an Auslander–Reiten sequence ending in .     

τ-Categories I: Ladders

In this series of papers, we introduce τ-categories, which are additive categories with some kind of Auslander–Reiten sequences. We apply them to study the category of lattices over orders. In this first paper, we study minimal projective resolutions in functor categories over τ-categories. Then we give a structure theorem of completely graded τ-categories using mesh categories. Presented by K. Roggenkamp Mathematics Subject Classifications (2000) primary 16G30; secondary 16E65, 16G70, 18E05. Osamu Iyama: Current address: Department of Mathematics, University of Hyogo, Himeji, 671-2201, Japan. e-mail: iyama@sci.u-hyogo.ac.jp.     

Regular modules and quasi-lengths over a 3-Kronecker quiver: using Fibonacci numbers

Let Q be a 3-Kronecker quiver (i.e., two vertices and three arrows having the same starting and ending vertices). The dimension vectors of the indecomposable regular representations X such that |X| = |τ ⁱ X| will be studied using the Fibonacci numbers, where |X| denotes the length of X and τ denotes the Auslander–Reiten translation. The quasi-lengths of the indecomposable regular representations with dimension vectors (m, m) and (2m, m) will also be discussed.     

Auslander–Reiten Components Containing Cones

Let A be a locally finite Abelian R-category with Auslander–Reiten sequences and with Auslander–Reiten quiver (A). We give a criterion for Auslander–Reiten components to contain a cone and apply this result to various categories.     

The Generalized Auslander–Reiten Condition for the bounded derived category

We consider the bounded derived category D ^b (R mod) of a left Noetherian ring R. We give a version of the Generalized Auslander–Reiten Condition for D ^b (R mod) that is equivalent to the classical statement for the module category and is preserved under derived equivalences.     

Mutation in triangulated categories and rigid Cohen–Macaulay modules

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings. Dedicated to Professor Idun Reiten on the occasion of her 65th birthday     

Cohen–Macaulay Isolated Singularities with a Dualizing Module

We generalize results of Foxby concerning a commutative Nötherian ring to a certain noncommutative Nötherian algebra Λ over a commutative Gorenstein complete local ring. We assume that Λ is a Cohen–Macaulay isolated singularity having a dualizing module. Then the same method as in the commutative cases works and we obtain a category equivalence between two subcategories of mod Λ, one of which includes all finitely generated modules of finite Gorenstein dimension. We give examples of such algebras which are not Gorenstien; orders related to almost Bass orders and some k-Gorenstein algebras for an integer k.Presented by I. Reiten ^★The author is supported by Grant-in-Aid for Scientific Researches B(1) No. 14340007 in Japan.     

τ-Categories III: Auslander Orders and Auslander–Reiten Quivers

We study the relationship between representation theoretic properties and homological properties of orders. We show that there is a close relationship among Auslander orders, τ-categories and Auslander regular rings. As an application, we give a combinatorial characterization of finite Auslander–Reiten quivers of orders. Presented by K. Roggenkamp Mathematics Subject Classifications (2000) Primary: 16G30; secondary: 16E65, 16G70, 18E05. Current address: Department of Mathematics, University of Hyogo, Himeji, 671-2201, Japan. e-mail: iyama@sci.u-hyogo.ac.jp     

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that , where is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formf_M: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G60, 16G70, 16G20.