Auslander-Reiten Quiver Research Articles

Approximations with modules having linear resolutions

Let Λ be a Koszul algebra over a field K. We study in this paper a class of modules closely related to the Koszul modules called weakly Koszul modules. It turns out that these modules have some special filtrations with modules having linear resolutions and therefore easy to describe minimal projective resolutions. We prove that if the Koszul dual of a finite-dimensional Koszul algebra is Noetherian then every finitely generated graded module has a weakly Koszul syzygy and as a consequence a rational Poincaré series. If Λ is selfinjective Koszul, we prove that the stable part of each connected component of the graded Auslander–Reiten quiver containing a weakly Koszul module is of the form ZA∞, and if the Koszul dual of Λ is Noetherian, then every component has its stable part of the form ZA∞.

Embedding of the vertices of the Auslander–Reiten quiver of an iterated tilted algebra of Dynkin type Δ in [formula omitted

Let Δ be a Dynkin diagram and k an algebraically closed field. Let A be an iterated tilted finite-dimensional k-algebra of type Δ and denote by A ̂ its repetitive algebra. We approach the problem of finding a combinatorial algorithm giving the embedding of the vertices of the Auslander–Reiten quiver Γ A of A in the Auslander–Reiten quiver Γ( mod ( A ̂ ))≃ ZΔ of the stable category mod ( A ̂ ) . Let T be a trivial extension of finite representation type and Cartan class Δ. Assume that we know the vertices of ZΔ corresponding to the radicals of the indecomposable projective T-modules. We determine the embedding of Γ A in ZΔ for any algebra A such that T( A)≃ T.

Weakly shod algebras

We study the class of algebras having a path of nonisomorphisms from an indecomposable injective module to an indecomposable projective module and any such path is bounded by a fixed number. We show that such an algebra is an iteration of one-point extensions starting at a product of tilted algebras. This allows us to describe, for instance, its Auslander–Reiten quiver.

Preprojective Modules and Auslander-Reiten Components

Auslander and Smalø (Auslander, M., Smalø, S. O. (1980). Preprojective modules over artin algebras. J. Algebra 66:61–122) introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. Coelho (Coelho, F. U. (1993). Components of Auslander-Reiten quivers with only preprojective modules. J. Algebra 157: 472–488) studied Auslander-Reiten components containing only hereditarily preprojective modules and gave a number of characterizations of such components. We shall study further these modules by using the description of shapes of semi-stable Auslander-Reitan components; see Liu (Liu, S. (1992). Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc. 45(2):32–54; Liu, S. (1993). Semi-stable components of an Auslander-Reiten quiver. J. London Math. Soc. 47(2):405–416). Our results will imply the result of Coelho (Coelho, F. U. (1993). Components of Auslander-Reiten quivers with only preprojective modules. J. Algebra 157: 472–488, (1.2)) and that of Auslander-Smalø (Auslander, M. and Smalø, S. O. (1980). Preprojective modules over artin algebras, J. Algebra 66:61–122, (9.16)). As an application, moreover, we shall show that a stable Auslander-Reiten component with “few” stable maps in TrD-direction is of shape ℤA ∞.

Dimension of the Mesh Algebra of a Finite Auslander–Reiten Quiver

Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, Bongartz and Gabriel (Bongartz, K., Gabriel, P., (1982). Covering spaces in representation theory. Invent. Math. 65:331–378.). A translation quiver defines a mesh algebra over any field. A natural question arises as to whether or not the dimension of the mesh algebra depends on the field. The purpose of this note is to show that the dimension of the mesh algebra of a finite Auslander–Reiten quiver over a field is a purely combinatorial invariant of this quiver. Indeed, our proof yields a combinatorial algorithm for computing this dimension. As a further application, one may use then semicontinuity of Hochschild cohomology of algebras as in Buchweitz and Liu (Buchweitz, R.-O., Lui, S. Hochschild cohomology and representation-finite algebras. Preprint.) to conclude that a finite Auslander–Reiten quiver contains no oriented cycle if its mesh algebra over some field admits no outer derivation.

Domestic Algebras with Many Nonperiodic Auslander–Reiten Components

We exhibit a wide class of domestic finite dimensional algebras over an algebraically closed field whose Auslander–Reiten quivers admit infinitely many connected components of type ℤ𝔻∞ (respectively, of types ℤ𝔻∞ and ℤ). The algebras are suitable iterated one-point extensions of hereditary algebras of Euclidean type 𝔸˜ n .

On Cohen—Macaulay Modules on Surface Singularities

Dedicated to V. I. Arnold Abstract. We study Cohen-Macaulay modules over normal surface singularities. Using the method of Kahn and extending it to families of modules, we classify Cohen-Macaulay modules over cusp singularities and prove that a minimally elliptic singularity is Cohen-Macaulay tame if and only if it is either simple elliptic or cusp. As a corollary, we obtain a classification of Cohen-Macaulay modules over log-canonical surface singularities and hypersurface singularities of type Tpqr; especially they are Cohen-Macaulay tame. We also calculate the Auslander-Reiten quiver of the category of Cohen-Macaulay modules in the considered cases. 2000 Math. Subj. Class. Primary 13C14, 13C05; Secondary: 16G50, 14J17.

On local right pure semisimple rings of length two or three

We investigate in this paper non-commutative local rings of the smallest length that are potential counter-examples to the pure semisimplicity conjecture. Throughout the paper is an associative ring with an identity element. We call local, if the Jacobson radical ( ) of is a two-sided maximal ideal. We denote by mod( ) the category of finitely generated right -modules. Given a right -module of finite length we denote by ( ) the length of . We recall that a ring is said to be of finite representation type, if is artinian and the number of the isomorphism classes of finitely generated indecomposable right (and left) -modules is finite. Following [24] we call a ring right pure semisimple, if every right -module is a direct sum of finitely generated modules. It is well known that a ring is of finite representation type if and only if is right pure semisimple and is left pure semisimple (see [2], [11], [18], [22]–[24]). It is still an open question, called the pure semisimplicity conjecture, if a right pure semisimple ring is of finite representation type (see [2] and [24], [25], [28]). In [13] the question is answered in affirmative for rings satisfying a polynomial identity and for self-injective rings (see also [7], [19] and [31]). The reader is referred to [42] and to the author’s expository papers [30] and [32] for a basic background and historical comments on the pure semisimplicity conjecture. It was shown by the author in [28] and [33] that there is a chance to find a counter-example to the pure semisimplicity conjecture and might be hereditary with two simple non-isomorphic modules. The existence of a counter-example depends on a generalized Artin problem on division ring extensions. In the present paper we are mainly interested in the existence of counter-examples to pure semisimplicity conjecture that are local of the smallest length, that is, of length ( ) two or three. This continues our study started in [28], [35] and [33]. It is shown in Lemma 3.1 that every such a local ring has ( )2 = 0. Therefore we study representation-infinite right pure semisimple local rings with ( )2 = 0 such that the Auslander-Reiten quiver (mod ) is of the form · · · → •→ •→ •→

Relations for the Grothendieck groups of triangulated categories

A class of triangulated categories with a finiteness condition is singled out. These triangulated categories have Auslander–Reiten triangles. It is proved that the relations of the Grothendieck group of a triangulated category in this class are generated by all Auslander–Reiten triangles. Moreover, the Auslander–Reiten quivers of certain triangulated categories in this class are described in terms of Dynkin diagrams.

Finite-Dimensional Representations of a Quantum Double

Let k be a field and let An(ω) be the Taft's n2-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a ribbon Hopf algebra. In a previous paper we constructed an n4-dimensional Hopf algebra Hn(p, q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω− 1, and studied the irreducible representations of Hn(1, q). We continue our study of Hn(p, q), and we examine the finite-dimensional representations of H3(1, q), equivalently, of D(A3(ω)). We investigate the indecomposable left H3(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H3(1, q), and the Auslander–Reiten quiver of H3(1, q).

Locations of modules for Brauer tree algebras

Suppose Λ is a Brauer tree algebra. We determine the location of a Λ-module M in the stable Auslander–Reiten quiver of Λ from the description of M as a multi-pushout of elementary modules. This is done by introducing a new combinatorial object associated to M, which is a certain walk around the Brauer tree of Λ. These walks have applications to determining universal deformation rings and stable homomorphism groups.

LOCALLY DYNKIN QUIVERS AND HEREDITARY COALGEBRAS WHOSE LEFT COMODULES ARE DIRECT SUMS OF FINITE DIMENSIONAL COMODULES

ABSTRACT Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We prove that every left C-comodule is a direct sum of finite dimensional C-comodules if and only if C is comodule Morita equivalent (see [19]) with a path K-coalgebra , where Q is a pure semisimple locally Dynkin quiver, that is, Q is either a finite quiver whose underlying graph is any of the Dynkin diagrams , , , , , , , or Q is any of the infinite quivers , , , with , shown in Sec. 2. In particular, we get in Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriel's theorem [11] characterising representation-finite hereditary K-algebras (see also [[6], Sec. VIII.5]). It is shown in Sec. 3 that if , then the Auslander-Reiten quiver of the category of finite dimensional left comodules has at most four connected components, and is connected if and only if Q has no sink vertices and .

ON NON-SEMIREGULAR COMPONENTS CONTAINING PATHS FROM INJECTIVE TO PROJECTIVE MODULES

ABSTRACT Let be a component of the Auslander-Reiten quiver of an Artin algebra containing projective and injective modules. Assume that the length of any path from an injective in to a projective in is bounded by some fixed number. We prove here that such component has no oriented cycles and is generalized standard, so containing only finitely many -orbits. Components of the Auslander-Reiten quiver of an Artin algebra containing paths in ind from injective to projective modules have appeared naturally in some classes of algebras such as quasitilted[7] or shod.[4] In both cases, these paths can be refined to paths of irreducible maps and any such path has either none hooks (in the case of a quasitilted algebra) or at most two of them (in the case of a shod algebra). See below for definitions.Here, we are interested in studying the components of such that there exists a number such that any path from an injective to a projective lying on it has at most hooks. We shall see that this is equivalent to the existence of a number such that any path in ind from an injective to a projective lying on it has length at most (Theorem 4.1). Such a component does not have oriented cycles (Corollary 3.4) and it is generalized standard (Theorem 4.3), hence containing only finitely many -orbits.In fact, when considering the existence of oriented cycles in such components, we shall prove the following more general result. If is a component of such that the number of hooks in any path of irreducible maps from an injective in to a projective in is bounded, then has no oriented cycles (Theorem 3.1). This generalizes Liu's main result in,[11] where he shows that any component of such that any path of irreducible maps from an injective in to a projective in is sectional (that is, with no hooks) has no oriented cycles. Observe that Li[8] has also considered such components and has shown that the above condition on the paths from injectives to projectives characterizes the existence of a section in . The proof of Theorem 3.1, however, follows closely some ideas contained in[6], also generalizing results proven there for quasitilted algebras. In a forthcoming paper,[5] we shall make use of the results proven here to study the class of algebras such that the length of any path from an indecomposable injective module to an indecomposable projective module is bounded by some number . This class of algebras contains the quasitilted and the shod algebras. These results were proven in an exchange project between Brazil and Uruguay. Both authors acknowledge financial support given by CNPq and FAPESP, Brazil and PEDECIBA, Uruguay.

Auslander–Reiten Components Containing Cones

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

Infinite Components of An Auslander-Reiten Quiver with Finite DTr-Orbits

Let A be a basic connected finite dimensional algebra over an algebraically closed field. We show that if Γ is an infinite connected component of the Auslander-Reiten quiver ΓA of A in which each ΓA-orbit contains only finitely many vertices, then the number of indecomposable direct summands of the middle term of any mesh, whose starting vertex belongs to the infinite stable part of Γ, is less than or equal to 3. Moreover, if the nonstable vertices belong to τA-orbits of exceptional projectives in Γ, then Γ can be obtained from a stable tube by a finite number of multiple coray-ray insertions of type α*γ and multiple coray-ray insertions of type α*γ.

Relative Auslander–Reiten sequences for quasi-hereditary algebras

Let A be a finite-dimensional algebra which is quasi-hereditary with re- spect to the poset (�,�), with standard modules �(�) for 2 �. Let F(�) be the category of A-modules which have filtrations where the quotients are standard modules. We determine some inductive results on the relative Auslander-Reiten quiver of F(�). Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in the study of highest weight categories arising in the representation theory of semisimple Lie algebras and algebraic groups (2), and they were studied by Dlab and Ringel and others (4, 5). LetA be quasi-hereditary. Then the simple A-modules are labelled as L(�) forin some partially ordered set (�,≤). Of central importance are the standard modules, denoted by �(�), and the categoryF(�) of modules which have a filtration where the quotients are standard modules. In (12) it was proved that this category has relative Auslander-Reiten (AR) sequences. The aim of our paper is to present some inductive results on the relative Auslander-Reiten quiver ofF(�). For any subset of �, letF (�) be the category of modules inF(�) where all quotients are �(�) with �∈ . If � is an ideal in the poset (�,≤) then there is a subalgebra A of A which is quasi-hereditary with poset (, ≤). In this case, the categoryF (�) is equivalent toF(�A ). A main result in this paper relates, for a non-projective indecomposable mod- ule X inF (�), the two relative Auslander-Reiten sequences with cokernel X inF(�) and inF (�) (Theorem 2.5). By Ringel duality, there is an equivalent version for categories of modules filtered by costandard mod- ules. We describe this and some variations in Section 3. We continue with further results on relative Auslander-Reiten sequences. In particular, we determine certain relative Auslander-Reiten sequences where a projective- injective summand occurs; this generalizes the usual almost split sequence where a projective-injective module occurs.

An algorithm for finding all preprojective components of the Auslander-Reiten quiver

The Auslander-Reiten quiver of a finite-dimensional associative algebra A encodes information about the indecomposable finite-dimensional representations of A and their homomorphisms. A component of the Auslander-Reiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the Auslander-Reiten translation. Preprojective components play an important role in the present research on algebras of finite and tame representation type. We present an algorithm which detects all preprojective components of a given algebra.

ON SMALL RIGHT PURE SEMISIMPLE RINGS AND THE STRUCTURE OF THEIR AUSLANDER-REITEN QUIVER

Let R be an associative ring with an identity element. One of the main aims of this paper is to show how potential counter-examples R to the pure semisimplicity conjecture of length ℓ(RR ) two or three can look like, and the shape of their Auslander-Reiten quiver is described. *Partially supported by Polish KBN Grant 2 P0 3A 012 16

On Auslander-Reiten components and simple modules for finite groups of Lie type

Let kG be the group algebra of a finite group G over an algebraically closed field k of characteristic p, where p is a prime. Let B be a p-block of kG with defect group AE(B) =G D. Then B is called to be of wild representation type, if D is neither cyclic, nor dihedral, semi-dihedral or a generalized quaternion group. Let 2 be a connected component of the stable Auslander-Reiten quiver 0s(B) of a wild p-block B of kG. In [6] K. Erdmann has shown that the tree class of 2 is A1. It is therefore natural to ask where a simple kG-module S of a wild p-block B can be found in its Auslander-Reiten component 2(S). The first author proved in [11] that a simple kG-module S of a wild p-block B must lie at the end of 2(S) if G is p-solvable. Moreover, he showed that, if there is some simple module which does not lie at the end, then there exist several simple modules lying at the end and having uniserial projective covers. We hope that this result may give a device for determining where simple modules lie in 0s(B). In this paper, we prove a similar assertion for wild p-blocks of a finite group G of Lie type, when p is the defining characteristic of G. Our main result is as follows. Theorem. Let G be a finite group of Lie type defined over a field K of characteristic p. Let B be a block of G with full defect of wild representation type. Then any simple kG-module S of B lies at the end of its Auslander-Reiten component 2(S).

COILS FOR VECTORSPACE CATEGORIES

Coils as components of Auslander–Reiten quivers of algebras and coil algebras are introduced by Assem and Skowroński. This concept is applied in the present paper to vectorspace categories. The four admissible operations on an Auslander–Reiten component of a vectorspace category, and the notions of v-coils and of vcoil vectorspace categories are introduced. A detailed study on the indecomposable objects of factorspace category of a vcoil vectorspace category is carried out.