Representation Theory Of Artin Algebras Research Articles

Exact categories, big Cohen-Macaulay modules and finite representation type

Abstract One of the first remarkable results in the representation theory of artin algebras , due to Auslander and Ringel-Tachikawa, is the characterisation of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce “big objects” and prove an Auslander-type “splitting-big-objects” theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.

The Jordan-Hölder property and Grothendieck monoids of exact categories

We investigate the Jordan-Hölder property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. We introduce Bruhat inversions of permutations and show that simples in a torsion-free class are in bijection with Bruhat inversions of the corresponding c-sortable element. We use this to give a combinatorial criterion for (JHP).

Duality and contravariant functors in the representation theory of artin algebras

We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring [Formula: see text] as the kernels of certain functors [Formula: see text] rather than of functors [Formula: see text] which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that [Formula: see text] is an artin algebra. Suppose that [Formula: see text] is an artin algebra. An additive functor [Formula: see text] preserves inverse limits and [Formula: see text] is finitely presented if and only if there is a sequence of natural transformations [Formula: see text] for some [Formula: see text] which is exact when evaluated at any left [Formula: see text]-module. Any additive functor [Formula: see text] with one of these equivalent properties has a definable kernel, and every definable subcategory of [Formula: see text] can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form [Formula: see text] for an artin algebra [Formula: see text]. That is, our results are extended to those locally finitely presented [Formula: see text]-linear categories whose finitely presented objects form a dualizing variety, where [Formula: see text] is a commutative artinian ring.

The Bongartz’s Theorem on the Homotopy Category

Let $$\Lambda $$ be an Artin algebra and $$M\in $$ mod $$\Lambda $$ . It is well-known that the Bongartz’s Theorem plays an important role in the representation theory of Artin algebras. In this paper, we investigate the relative version of the Bongartz’s Theorem in the bounded homotopy category $$K^{b}(\text {add}M)$$ and construct a Bongartz’s completion for an $$\text {add}M$$ -relative presilting complex in $$K^{b}(\text {add}M)$$ such that it becomes an $$\text {add}M$$ -relative silting complex. Finally, we give some applications.

Classifying exact categories via Wakamatsu tilting

Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can be described in terms of Wakamatsu tilting (= semi-dualizing) subcategories. If moreover the exact category has higher kernels, then its image coincides with the category naturally associated with a cotilting subcategory up to summands. We apply these results to the representation theory of artin algebras. In particular, we show that the ideal quotient of a module category by a functorially finite subcategory closed under submodules is a torsionfree class of some module category.

On the dimensions of path algebras

1. IntroductionThe notion of representation dimension of an artin algebra Λ, denoted rep.dimΛ, hasbeen introduced by Auslander in his Queen Mary Notes in 1971. His hope was to provide“a reasonable way of measuring how far the category of finitely generated modules over anartin algebra is from being of finite representation type”. Recall that an artin algebra Λis called of finite representation type if, up to isomorphisms, there are only finitely manyfinitely generated indecomposable Λ-modules. Such algebras are called representation finite.The importance of this class of artin algebras for the whole representation theory of artinalgebras is well understood, see e.g. [Au], [Ga] and [Ta].Auslander proved that Λ is of finite representation type if and only if its representationdimension is at most two. But it turned out to be hard to compute the actual value ofthe representation dimension of a given algebra. Moreover, all the algebras that one couldcompute explicitly their representation dimensions, had representation dimension at mostthree. So a natural question arises: are there artin algebras of representation dimensiongreater than three. Of course such algebras are of infinite representation type. On the otherhand, another question was whether this invariant is always finite. These questions has beenanswered positively, at almost the same time. Iyama [I] in 2003 proved that it is alwaysfinite and Rouquier [Rq1] in 2005 showed that any natural number n, except 1, can occuras the representation dimension of some algebra. In fact, he showed that the representationdimension of the exterior algebra Λk

An Auslander-type result for Gorenstein-projective modules

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105–179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269–310].

Auslander-Reiten sequences and global dimensions

1. IntroductionAlmost split sequences were introduced by Auslander and Reiten. Their significantimportance in the representation theory of Artin algebras and groups is well-knownin the literature. In [11] the relationship between projective dimensions of modulesappearing in an almost split sequence was studied. However, the relationship betweenthe global dimensions of endomorphism algebras of these modules is unknown. In thispaper we shall establish an inequality for global dimensions of these endomorphismalgebras. This reveals also that almost split sequences may play a role in the studyof representation dimensions of algebras, or the global dimensions of endomorphismalgebras of certain modules. Our main result isTheorem 1.1. Let Abe an Artin algebra, let M be a generator-cogenerator for A-mod, and let sbe a positive integer. Then:(1) If α

Quadratic bimodules and quadratic orders

In some branches of mathematics, especially the representation theory of Artin algebras and orders, it has been an important problem to study the (geometric) quotient GLn(Λ)\Mn(M) for a (Λ,Λ)-module M (g · m := gmg−1 for g ∈ GLn(Λ) and m ∈ Mn(M)). An effective formulation of such problems is a bimodule problem [4,9], which is a study of the matrix category Mat(C,M) associated to an additive category C and a (C,C)-module M (Section 2.1). The bimodule problems have nice module-theoretical interpretation by means of prinjective modules [22,29,30]. They have already many useful applications in the representation theory of finite-dimensional algebras, artinian rings, orders, vector space categories, and bocses, see [6,7,9,16,19,25,28,29]. In this paper, we introduce the concept of quadratic extensions of rings (Section 1.2), which are closely related to many rings known in representation theory, for example Bass orders [10,17], Backstrom orders [28], Green orders [27], special biserial algebras [31], and clannish algebras [5] (see Sections 1.3, 3.4, and 3.5). Using quadratic extensions, we introduce the concept of quadratic bimodules (Section 3.1), where we can easily formulate self-reproducing systems [21] and clans [5] in terms of quadratic bimodules. Then we study their matrix category Mat(C,M) associated to a quadratic (C,C)-module M , especially we give a description of all indecomposable objects in terms of walks (Sections 3.3, 6.1), which can be regarded as a generalization of Green walks [13] (Section 3.4), strings and bands [5,31] (Section 3.5). As an application, we give a description of finitely generated indecomposable modules over quadratic orders (Section 1.3(2)) in Section 7.3. In forthcoming papers, our results will be applied in construction of large new classes of orders of tame representation type. Some of the results of this paper were presented at the Conference and Workshop Constanta 2000 and were announced in [19] without proof.

On the Representation Dimension of Finite Dimensional Algebras

The importance of the representation finite Artin algebras for the whole representation theory of Artin algebras is well understood and there is much literature on the subject (see [1, 6, 12]). These are Artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules. Moreover there is a bijective correspondence between the class of representation finite Artin algebras and that of Artin algebras with global dimension of at most 2 and with dominant dimension of at least 2. This result was established by Auslander in [1]. Motivated by this correspondence, Auslander introduced the concept of representation dimension for Artin algebras to study the connection of arbitrary Artin algebras with representation finite Artin algebras. “It is hoped that this notion gives a reasonable way of measuring how far an Artin algebra is from being representation finite type” [1, p. 134]. Representation dimension is a Morita-invariant of Artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators. Unfortunately, little seems to be known about representation dimension. So there are a lot of essential questions on this invariant. Dealing with the computation and estimation of the representation dimension of an Artin algebra, there are a few cases known: Auslander proved that the representation dimension of an Artin algebra is two if and

Book Review: Representation theory of Artin algebras

Book Review: Representation theory of Artin algebras

Almost split sequences in dimension two

(a) (*) does not split. (b) If h: X-r C, is a morphism in C which is not a splittable epimorphism in C, then there is a t: X+ C, such that gt = h. (c) If j: Ci + Y is a morphism in C which is not a splittable monomorphism in C, then there is an s: C, + Y in C such that j = sf Almost split sequences were first introduced by us in the case A is the category of finitely generated modules over an artin algebra /i and C = A (see [6, 71). The reader is referred to the expository articles [ 11, 121 for an account of the role almost split sequences have played in the theory of the representation theory of artin algebras. That almost split sequences exist for various subcategories of module categories over a much wider class of rings than artin algebras was first shown in [2, 31, where various existence theorems for almost split sequences were given. These results seemed to indicate that there might be connections between the structure and existence of almost split sequences and algebraic geometry. Recent developments show that this is indeed the case. In [4] it is shown that the structure of the almost split sequences in the category of reflexive modules of a complete rational double point over the complex numbers determines the desingularization graph of the singularity. Also it has been shown in [S] that a complete Cohen-Macaulay local ring S is an isolated singularity if and only if the category of Cohen-Macaulay

An Introduction to the Representation Theory of Artin Algebras

Bulletin of the London Mathematical SocietyVolume 17, Issue 3 p. 209-233 Special article An Introduction to the Representation Theory of Artin Algebras Idun Reiten, Idun Reiten University of Trondheim, AVH Department of Mathematics, 7055 Dragvoll NorwaySearch for more papers by this author Idun Reiten, Idun Reiten University of Trondheim, AVH Department of Mathematics, 7055 Dragvoll NorwaySearch for more papers by this author First published: May 1985 https://doi.org/10.1112/blms/17.3.209Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume17, Issue3May 1985Pages 209-233 RelatedInformation

Skew group algebras in the representation theory of artin algebras

Then the corresponding crossed product algebra /i * ?G, or /i x G for short, has an elements CgiEG ,gi A.-; li ~/i. Addition is componentwise, and multiplication is given by & = g(A)g and g, g2 = y(g, , g2) m. In this paper we assume that the values of y lie in the center Z(A) of/i. Hence the action of G on /i is given by a group homomorphism G-P Aut(/i), and (3) can be left out. In the special case that y is the trivial map we write /1G instead of/i * G, and the elements as C,,,c li gi. AG is then called a skew group ring. There is a lot of literature on skew group algebras and crossed product algebras, and on the relationship with the ring AC whose elements are those elements of A left fixed by G. Much work has been done on which properties of li are inherited by n * G or AC. Some of the work on the relationship between these rings has its roots in trying to develop a Galois theory for noncommutative rings. We refer to [3, 7, 13-15, 19, 21, 23-25, 27, 281 and their references. In this paper we study these constructions when /i is an artin algebra and G a finite group of order n such that n is invertible in /1. Under these assumptions the construction preserves central properties of interest in the

The Inductive Step of the Second Brauer-Thrall Conjecture

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.Section 1 is devoted to giving the necessary background in the theory of almost split sequences.As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].

Representation theory of artin algebras v: methods for computing aimost split sequences and irreducible morphisms

(1977). Representation theory of artin algebras v: methods for computing aimost split sequences and irreducible morphisms. Communications in Algebra: Vol. 5, No. 5, pp. 519-554.

Representation theory of artin algebras IV1)Invariants given by almost split sequences

(1977). Representation theory of artin algebras IV1)Invariants given by almost split sequences. Communications in Algebra: Vol. 5, No. 5, pp. 443-518.

The Equivalence of Certain Functors Occurring in the Representation Theory of Artin Algebras and Species

The Equivalence of Certain Functors Occurring in the Representation Theory of Artin Algebras and Species

Representation theory of artin algebras iii almost split sequences

(1975). Representation theory of artin algebras iii almost split sequences. Communications in Algebra: Vol. 3, No. 3, pp. 239-294.