Algèbre/Algebra

Dualizing rings and a characterization of finite representation type

Henning Krause

Abstract: Using the relation between indecomposable endofinite right and left Λ-modules, we introduce (right) dualizing rings and show that a ring Λ is of finite representation type if and only if Λ is right dualizing and every right Λ-module is a direct sum of (finitely presented) indecomposable Λ-modules. We present also criteria for the existence of non-finitely presented indecomposable endofinite modules over a dualizing ring.

Anneaux dualisants: Une charactérisation des anneaux de répresentation finie

Resumé: Nous introduisons la notion d’anneau dualisant (à droite) en usant de la relation entre modules indécomposable endofinis à droite et à gauche. Nous montrons qu’un anneau Λ est de répresentation finie si et seulement si Λ est dualisant à droite est que tout Λ-module à droite est somme directe de Λ-modules indécomposables (de présentation finie). Nous exhibons en outre des critères d’existence de modules indécomposable endofinis de présentation infinie sur un anneau dualisant.

Version française abrégée: Soit Λ un anneau de répresentation finie, c’est-à-dire un anneau artinien à droite qui n’admet à isomorphisme près qu’un nombre fini de modules à droite indécomposables et de présentation finie. Auslander [1] et Ringel/Tachikawa [7] ont montré dans ce cas que tout Λ-module à droite est somme directe de Λ-modules de présentation finie. Nous démonstrons ici la réciproque pour les anneaux Λ vérifiant la condition suivante:

(D) Pour tout Λ-module à droite indécomposable, endofini et de présentation fini, le dual est un Λ-module à gauche de présentation finie (donc indécomposable et endofini).

Rappelons qu’un Λ-module à droite M est endofini s’il est de longueur finie comme module sur Γ = End_{Λ}(M)^{op}. Lorsque M est indécomposable, de présentation finie et endofini sur Λ, nous appelons dual de M le Λ-module à gauche DM = Hom_{Γ}(M, I), où I est une enveloppe injective du seul Γ-module simple. Nous appelons Λ dualisant à droite lorsque la condition (D) est remplie. Lorsque Λ et Λ^{op} vérifient (D), nous appelons Λ dualisante. Voici quelques exemples:

(1) Tout anneau Λ de représentation finie est dualisant car Λ^{op} est également de représentation finie.

(2) Toute algèbre noethérienne, c’est-à-dire tout anneau dont le centre est noethérien et qui est un module de type fini sur ce centre, est dualisant [3, Proposition 7.1].

(3) Tout anneau Λ artinien à gauche et à droite tel que Λ/ rad Λ soit un module de type fini sur son centre, est dualisant [8].

Ces examples montrent que le théorème suivant généralise des énconcés de Auslander [2] et Herzog [5].

Théorème Pour tout anneau Λ, les conditions suivantes sont équivalentes:

(1) Λ est de représentation finie.

(2) Λ est dualisant à droite et tout Λ-module à droite est somme directe de Λ-modules de présentation finie.

La démonstration de cet énconcé repose sur une analyse de la catégorie (mod(Λ^{op}), Ab) des foncteurs additifs de la catégorie des Λ^{op}-modules a droite de présentation finie dans celle des groupes abéliens. Nous utilisons également la notion de localisation d’une catégorie de Grothendieck telle qu’on peut la trouver dans [4]. Comme nous l’a signalé M. Prest, que nous remercions, une alternative de démonstration utilise des idées de [5].

Let Λ be a ring of finite representation type, i.e. Λ is right artinian and there is only a finite number of finitely presented indecomposable right Λ-modules. Then it has been shown by Auslander [1] and by Ringel and Tachikawa [7] that every right Λ-module is a direct sum of finitely presented indecomposable Λ-modules. In this note we shall prove the converse for rings Λ which satisfy the following condition:

(D) If M is a finitely presented indecomposable endofinite right Λ-module, then its dual DM is a finitely presented indecomposable endofinite left Λ-module.

Recall that a right Λ-module M is endofinite if it is of finite length when regarded in the natural way as a Γ = End_{Λ}(M)^{op}-module. If M is finitely presented indecomposable and endofinite, we denote by DM the left Λ-module Hom_{Γ}(M, I), where I is an injective envelope of the unique simple Γ-module. Note that DM is automatically indecomposable and endofinite. We call Λ right dualizing if the condition (D) is satisfied. If Λ and Λ^{op} both satisfy (D), we call Λ dualizing. We include some examples.

(1) Any ring Λ of finite representation type is dualizing since Λ^{op} is also of finite representation type.

(2) If Λ is a noetherian algebra, i.e. the center Z(Λ) is noetherian and Λ is a finitely generated Z(Λ)-module, then Λ is dualizing [3, Proposition 7.1].

(3) Let Λ be a two-sided artinian ring. If Λ is a PI-ring or, equivalently, if Λ/ rad Λ is an artin algebra, then Λ is dualizing [8].

These examples show that the following theorem generalizes results of Auslander [2] and Herzog [5].

Theorem For a ring Λ the following conditions are equivalent:

(1) Λ is of finite representation type.

(2) Λ is right dualizing and every right Λ-module is a direct sum of (finitely presented) indecomposable Λ-modules.

The proof of this result is based on an analysis of the category (mod(Λ^{op}), Ab) of all additive functors from the category of finitely presented right Λ^{op}-modules to the category Ab of abelian groups, and we shall assume that the reader is familiar with the localization theory for such a Grothendieck category which may be found in [4]. The author would like to thank M. Prest for pointing out that an alternative proof can be obtained by using ideas from [5].

Throughout we fix a ring Λ and we denote by Mod(Λ) the category of (right) Λ-modules. The fully faithful functor

Mod(Λ) → D(Λ) = (mod(Λ^{op}), Ab), M ↦ M ⊗_{Λ} −

is fundamental for our approach. It identifies the pure-injective Λ-modules with the injective objects in D(Λ). The category D(Λ) is locally coherent, i.e. the subcategory C(Λ) = fp(mod(Λ^{op}), Ab) of finitely presented functors is abelian, and we can reformulate the conditions of the theorem as follows.

Lemma 1 (1) Λ is of finite representation type iff D(Λ) is locally finite, i.e. every object in C(Λ) is of finite length.

(2) Every Λ-module is a direct sum of (finitely presented) indecomposable Λ-modules iff D(Λ) is locally noetherian, i.e. every object in C(Λ) is noetherian.

Proof : (1) is shown in [1] and (2) follows from the well-known fact that a functor category is locally noetherian iff every injective object decomposes into a direct sum of indecomposable objects.

Remark A ring Λ is said to be right pure-semisimple provided that every right Λ-module is pure-injective. An equivalent property is that D(Λ) is locally noetherian.

To use the condition (D) we recall the following result which is due to Crawley-Boevey.

Lemma 2 Let M be a finitely presented indecomposable endofinite Λ-module. If DM is finitely presented, then M ⊗_{Λ} − is the injective envelope of a simple functor in D(Λ).

Proof : The assumption implies that M ≅ DDM by [3, Proposition 6.3]. Thus M ⊗_{Λ} − is the injective envelope of a simple functor by [3, Theorem 2.3].

Proof of the theorem: We need to show that (2) implies (1). Therefore assume (2). Thus D(Λ) is locally noetherian by Lemma 1 and therefore its Krull dimension Kdim D(Λ) in the sense of Gabriel exists [4]. Recall, that the Krull filtration

0 = D(Λ)_{−1} ⊆ D(Λ)_{0} ⊆ … ⊆ D(Λ)_{α} ⊆ … ⊆ D(Λ)

consists of localizing subcategories of D(Λ) indexed by the ordinals, and Kdim D(Λ) denotes the least ordinal α such that D(Λ)_{α} = D(Λ). Moreover, Kdim D(Λ) = β +1 for some ordinal β since for any localizing subcategory C of D(Λ) we have C = D(Λ) iff Hom_{Λ^{op}}(Λ, ) ∈ C. We can choose an indecomposable injective object N from the quotient category D(Λ)/D(Λ)_{β} and denote by s_{β}(N) = M ⊗_{Λ} − the corresponding indecomposable injective object under the section functor s_{β}: D(Λ)/D(Λ)_{β} → D(Λ). We claim that M is endofinite. In fact, the section functor is a right adjoint of the quotient functor q_{β}: D(Λ) → D(Λ)/D(Λ)_{β} and this gives an isomorphism

M = Hom(Hom_{Λ^{op}}(Λ, ), s_{β}(N)) ≅ Hom(q_{β}(Hom_{Λ^{op}}(Λ, )), N)

of Γ-modules where Γ = End_{Λ}(M)^{op} ≅ End(N)^{op}. The claim follows from the next lemma, using the fact that X = q_{β}(Hom_{Λ^{op}}(Λ, )) is finitely presented and therefore of finite length since D(Λ)/D(Λ)_{β} is locally finite according to our construction.

Lemma 3 Let X and N be objects such that X is of finite length and N is injective. Then the End(N)^{op}-module Hom(X, N) is of finite length.

Proof : Any finitely generated End(N)^{op}-submodule of Hom(X, N) has the form Hom(X/U, N) for some subobject U of X since N is injective. In fact, one takes U = ∩_{i=1}^{n} Ker(φ_{i}) if φ_{1}, …, φ_{n} are generators of the submodule. Now the finite length of X implies that Hom(X, N) has both chain conditions.

Having shown that M is endofinite, we can use Lemma 2 as follows. The module M is indecomposable and therefore finitely presented by our assumptions. Thus M ⊗_{Λ} − has a simple subobject and therefore Hom(D(Λ)_{α}, M ⊗_{Λ} −) ≠ 0 for all α ≥ 0 by definition of the Krull filtration of D(Λ). On the other hand, Hom(D(Λ)_{β}, M ⊗_{Λ} −) = 0 and therefore β = −1. We conclude that Kdim D(Λ) = 0. Thus D(Λ) is locally finite and Λ is of finite representation type by Lemma 1. This finishes the proof of the theorem.

The preceding proof is based on ideas which allow us to deduce criteria for the existence of generic modules over dualizing rings. Here we call a module generic, if it is indecomposable and endofinite but not finitely presented. Note that the existence of a generic module is crucial for the validity of the second Brauer-Thrall conjecture for artin algebras [3].

Two definitions are needed for the statement of our result. An additve subgroup of a Λmodule M is called of finite definition if it arises as the kernel of a morphism M → M ⊗_{Λ} X, m ↦ m⊗x for some finitely presented Λ^{op}-module X, and some element x ∈ X. We mention the well-known fact that M is Σ-pure-injective iff M has the descending chain condition on subgroups of finite definition, and M is endofinite iff M has the descending and the ascending chain condition on subgroups of finite definition. Given a partially ordered set, a dense chain is a subset isomorphic to {m · 2^{−n} ∈ [0, 1] | m ∈ ℤ, n ∈ ℕ} with its usual ordering.

Theorem Let Λ be a right dualizing ring. Then there exists a generic Λ-module provided that one of the following conditions holds.

(1) There is a non-zero Λ-module having no finitely presented indecomposable endofinite direct summand and no dense chain of subgroups of finite definition.

(2) For some natural number n there is an infinite number of finitely presented indecomposable Λ-modules of endolength at most n.

A proof of this result will appear in [6]. For artin algebras, the second criterion is due to Crawley-Boevey [3].

References

[1] M. Auslander, Representation theory of artin algebras II, Comm. Algebra 1 (1974) 269–310.

[2] M. Auslander, Large modules over artin algebras, in: Algebra, Topology and Categories, Academic Press, 1976.

[3] W. Crawley-Boevey, Modules of finite endolength over their endomorphism ring, in: Representations of algebras and related topics, eds. S. Brenner and H. Tachikawa, London Math. Soc. Lec. Note Series 168 (1992) 127–184.

[4] P. Gabriel, Des catégories abéliennes, Bull. Soc. math. France, 90 (1962), 323–448.

[5] I. Herzog, A test for finite representation type, preprint (1993).

[6] H. Krause, Generic modules over artin algebras, preprint (1995).

[7] C.M. Ringel, H. Tachikawa, QF-3 rings, J. Reine Angew. Math. 272 (1975), 49–72.

[8] M. Schmidmeier, Diss. Univ. München, in preparation.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de