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.     

Auslander–Reiten components with bounded short cycles

We study Auslander–Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First, we give a number of combinatorial characterizations of almost acyclic Auslander–Reiten components. Then, we shall show that an Auslander–Reiten component with bounded short cycles is obtained, roughly speaking, by gluing the connecting components of finitely many tilted quotient algebras. In particular, the number of such components is finite and each of them is almost acyclic with only finitely many DTr-orbits. As an application, we show that an artin algebra is representation-finite if and only if its module category has bounded short cycles. This includes a well known result of Ringel’s, saying that a representation-directed algebra is representation-finite.     

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

We study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.

     

On Auslander–Reiten Components and Heller Lattices for Integral Group Rings

Let be a finite group, a complete discrete valuation ring of characteristic zero with residue class field of characteristic , and a block of the group ring . Suppose that is of infinite representation type and is sufficiently large to satisfy certain conditions. Let be the Auslander–Reiten quiver of and a connected component of . In this paper, we show that if contains some Heller lattices then the tree class of the stable part of is . Also, we show that has infinitely many components of type if a defect group of is neither cyclic nor a Klein four group.Presented by Jon Carlson.     

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 .     

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.     

Tilting complexes and Auslander–Reiten conjecture

We studied the properties of tilting complexes and proved that derived equivalences preserve the validity of the Auslander–Reiten conjecture.     

Auslander–Reiten Triangles in Homotopy Categories

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

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     

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.     

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of \(\operatorname{Spec}R\) , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.     

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     

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.     

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.