Constructing large modules over artin algebras

Henning Krause

Let Λ be an artin algebra and denote by mod(Λ) the category of finitely generated Λ-modules. Apart from those we are also interested in Λ-modules which are not finitely generated. These are called large. Recall that the algebra Λ is of infinite representation type provided that mod(Λ) has infinitely many isomorphism classes of indecomposable objects. A theorem of Auslander asserts that there exists a large indecomposable Λ-module whenever Λ is of infinite representation type [2]. In this note we use his method to construct specific large indecomposable modules which arise naturally from certain infinite families of morphisms in mod(Λ) having isomorphic kernels. We apply this result as follows. Suppose there is given a chain

X_{1} ⟶^{ψ_{1}} X_{2} ⟶^{ψ_{2}} … ⟶^{ψ_{s−1}} X_{s} ⟶^{ψ_{s}} …

of monomorphisms between finitely generated indecomposable Λ-modules. Denote by X_{∞} = ⟶ lim X_{i} the corresponding direct limit. It is natural to ask for conditions that ensure the indecomposability of X_{∞}. In fact, chains of monomorphisms of the above form occur for quasi-serial components of the Auslander-Reiten quiver of Λ. Ringel calls any module X = X_{s} belonging to such a component quasi-serial of quasi-length s, since it determines, up to isomorphism, uniquely a chain of irreducible monomorphisms ψ_{i}: X_{i} → X_{i+1}, i ∈ ℕ, such that there exists no irreducible monomorphism ending in X_{1} [8]. We show that the corresponding direct limit X_{∞} = ⟶ lim X_{i} is a large indecomposable module. Moreover, we obtain an exact sequence

0 ⟶ X ⟶^{ψ_{X}} X_{∞} ⟶ (Tr D)^{s}(X)_{∞} ⟶ 0

where (Tr D)^{s}(X) denotes the s-th power of the transpose of the dual of X which is again a quasi-serial module. This exact sequence has the following property. Every morphism θ: X → Y in mod(Λ) belongs to the infinite radical of mod(Λ) if and only if θ factors through ψ_{X}, and moreover, θ induces a commutative diagram of the following form

0 ⟶ X ⟶^{ψ_{X}} X_{∞} ⟶ (Tr D)^{s}(X)_{∞} ⟶ 0
↓θ  ↓  ↓
0 ⟶ Y ⟶^{ψ_{Y}} Y_{∞} ⟶ (Tr D)^{t}(Y)_{∞} ⟶ 0

provided that Y is quasi-serial of quasi-length t.

We would like to point out as an application that modules of the form X_{∞} can be used to construct generic modules in the sense of Crawley-Boevey [5]. This will be discussed in a forthcoming publication. Note that the existence of a generic module is crucial for the validity of the second Brauer-Thrall conjecture for artin algebras [5].

The construction

Let Λ be an artin algebra and denote by Mod(Λ) the category of right Λ-modules. Given two morphisms φ_{i}: X → Y_{i}, i = 1, 2, in Mod(Λ) we define φ_{2} ⪯ φ_{1} if there exists a morphism ψ: Y_{1} → Y_{2} such that φ_{2} = ψ∘φ_{1}. We write φ_{2} ≺ φ_{1} if φ_{2} ⪯ φ_{1} and φ_{1} ⪯̸ φ_{2}. Recall that a morphism φ: X → Y in mod(Λ) is irreducible provided that φ is neither a split monomorphism nor a split epimorphism and if for any factorization φ = φ_{2}∘φ_{1} in mod(Λ) either φ_{1} is a split monomorphism or φ_{2} is a split epimorphism.

Theorem 1 Let 0 → X →^{φ_{i}} Y_{i} →^{χ_{i}} Z_{i} → 0, i ∈ ℕ, be a family of exact sequences in mod(Λ). Suppose that χ_{i} is irreducible and that φ_{i+1} ≺ φ_{i} for all i ∈ ℕ. Then there exists a family of morphisms ψ_{i}: Y_{i} → Y_{i+1} satisfying φ_{i+1} = ψ_{i}∘φ_{i} for all i ∈ ℕ such that Y = ⟶ lim Y_{i} is a large denumerably generated indecomposable module.

We postpone a proof and discuss some preliminaries. We shall use the category (mod(Λ), Ab) of additive functors from mod(Λ) to the category Ab of abelian groups. Note that for a pair of morphisms φ_{i}: X → Y_{i}, i = 1, 2, in mod(Λ) we have φ_{2} ⪯ φ_{1} iff Im(Hom_{Λ}(φ_{2}, )) ⊆ Im(Hom_{Λ}(φ_{1}, )) as subfunctors of Hom_{Λ}(X, ). We shall also use the fully faithful functor

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

which identifies the pure-injective Λ-modules with the injective objects in (mod(Λ^{op}), Ab). Note that Ext^{1}( , X ⊗_{Λ} −) vanishes on finitely presented functors for every Λ-module X. Moreover, every finitely generated Λ-module is automatically pure-injective. The following characterization of an irreducible morphism in mod(Λ) is needed.

Lemma 2 For a non-split exact sequence 0 → X →^{φ} Y →^{χ} Z → 0 in mod(Λ) the following are equivalent.

(1) χ is irreducible.

(2) φ ⪯ θ or θ ⪯ φ for each morphism θ: X → Y′ in mod(Λ).

(3) φ ⪯ θ or θ ⪯ φ for each morphism θ: X → Y′ in Mod(Λ).

Proof : (1) ⇔ (2) See [3, Proposition 2.7].

(2) ⇔ (3) One direction is trivial. Therefore suppose (2). We apply the well-known duality fp(mod(Λ), Ab) → fp(mod(Λ^{op}), Ab) between the categories of finitely presented functors which sends Hom_{Λ}(X, ) to X ⊗_{Λ} − (e.g. see [6]). It follows from this duality that (2) is equivalent to the property that any subobject G of X ⊗_{Λ} − in (mod(Λ^{op}), Ab) either contains or is contained in F = Ker(φ ⊗_{Λ} −). Taking G = Ker(θ ⊗_{Λ} −) for a morphism θ: X → Y′ we obtain either a morphism Y ⊗_{Λ} − → Y′ ⊗_{Λ} − if F ⊆ G since Ext^{1}(Z ⊗_{Λ} −, Y′ ⊗_{Λ} −) = 0, or we obtain a morphism Y′ ⊗_{Λ} − → Y ⊗_{Λ} − if G ⊆ F since Ext^{1}( , Y ⊗_{Λ} −) = 0. Thus we have φ ⪯ θ or θ ⪯ φ.

Let C be an abelian category and F: C → Ab be an additive functor. A non-zero element x ∈ F (X) is called minimal if for each φ: X → Y in C the property F (φ)(x) ≠ 0 implies that φ is a monomorphism. The following facts are easily verified.

Lemma 3 (1) If x ∈ F (X) is minimal, then X is indecomposable.

(2) If X is a noetherian object in C and x ∈ F (X) is a non-zero element, then there exists an epimorphism φ: X → Y such that F (φ)(x) ∈ F (Y) is minimal.

Proof : See [1, Lemma 3.2].

Proof of Theorem 1 : Let F = Hom_{Λ}(X, )/∩_{i∈ℕ}Im(Hom_{Λ}(φ_{i}, )) and denote by π: Hom_{Λ}(X, ) → F the canonical morphism in (mod(Λ), Ab). Using the preceding lemmas one constructs easily a family of morphisms μ_{i}: X → W_{i}, i ∈ ℕ, together with a family of natural numbers 1 = n_{0} < n_{1} < … such that

(i) π(μ_{i}) ∈ F (W_{i}) is minimal for all i ∈ ℕ;

(ii) φ_{n_{i}} ≺ μ_{i} ⪯ φ_{n_{i}−1} for all i ∈ ℕ. Applying (ii) we obtain the following chain of morphisms.

Y_{1} ⟶^{ψ_{1}} … ⟶^{ψ_{n_{1}−2}} Y_{n_{1}−1} ⟶ W_{1} ⟶ Y_{n_{1}} ⟶^{ψ_{n_{1}}} … ⟶^{ψ_{n_{2}−2}} Y_{n_{2}−1} ⟶ W_{2} ⟶ Y_{n_{2}} ⟶^{ψ_{n_{2}}} …

Taking the corresponding compositions in this chain we obtain morphisms ψ_{i}: Y_{i} → Y_{i+1} and ν_{i}: W_{i} → W_{i+1}, i ∈ ℕ, such that

(iii) φ_{i+1} = ψ_{i}∘φ_{i} and μ_{i+1} = ν_{i}∘μ_{i} for all i ∈ ℕ;

(iv) Y = ⟶ lim Y_{i} and W = ⟶ lim W_{i} are isomorphic. To show that Y is indecomposable we extend the functor F: mod(Λ) → Ab to a functor F′: Mod(Λ) → Ab which commutes with direct limits and denote by π′: Hom_{Λ}(X, ) → F′ the corresponding morphism between functors from Mod(Λ) to Ab. Using (i) it is not hard to check that the morphism μ = ⟶ lim μ_{i} has the property that π′(μ) ∈ F′(W) is minimal. We refer to the proof of [2, Theorem 1.5] for details and conclude that Y is indecomposable.

Our next aim is to discuss some properties of the morphism φ = ⟶ lim φ_{i}: X → Y which arises in Theorem 1. We will use the following general fact.

Proposition 4 Let φ_{i}: X → Y_{i}, i ∈ I, be a directed family of morphisms in Mod(Λ) and denote by φ = ⟶ lim φ_{i}: X → Y its direct limit. Suppose also that Y′ ∈ Mod(Λ) is pureinjective. Then a morphism θ: X → Y′ factors through φ if and only if θ factors through φ_{i} for all i ∈ I.

Proof : One direction is clear. Therefore suppose that θ factors through φ_{i} for all i ∈ I. We obtain for each i an exact commutative diagram

0 ⟶ F_{i} ⟶ X ⊗_{Λ} − ⟶ Y_{i} ⊗_{Λ} −
↓  ∥  ↓
0 ⟶ G ⟶ X ⊗_{Λ} − ⟶ Y′ ⊗_{Λ} −

and therefore also an exact commutative diagram

0 ⟶ ⟶ lim F_{i} ⟶ X ⊗_{Λ} − ⟶ Y ⊗_{Λ} −
↓  ∥
0 ⟶ G ⟶ X ⊗_{Λ} − ⟶ Y′ ⊗_{Λ} −

since direct limits in (mod(Λ^{op}), Ab) are exact. Using the injectivity of Y′ ⊗_{Λ} − there exists a morphism Y ⊗_{Λ} − → Y′ ⊗_{Λ} − making the above diagram commutative. Thus the assertion follows.

Corollary 5 Let 0 → X →^{φ_{i}} Y_{i} →^{χ_{i}} Z_{i} → 0, i ∈ I, be a directed family of exact sequences in mod(Λ) and suppose that χ_{i} is irreducible for all i ∈ I. Suppose also that θ: X → Y′ is a morphism in Mod(Λ) where Y′ is pure-injective. Then the direct limit φ = ⟶ lim φ_{i}: X → Y has the following properties.

(1) θ factors through φ if and only if θ factors through φ_{i} for all i ∈ I.

(2) Either θ factors through φ or φ factors through θ.

Proof : Combine Proposition 4 and Lemma 2.

Quasi-serial modules

Families of irreducible epimorphisms having isomorphic kernels arise quite often for artin algebras of infinite representation type. In fact one may ask whether any artin algebra of infinite representation type admits such a family. We devote the second half of this paper to a class of finitely generated modules where such families naturally occur. To this end recall the following concept which was introduced in [8]. A component C of the Auslander-Reiten quiver of Λ is called quasi-serial if

(i) C contains neither projective nor injective modules;

(ii) if 0 → X → ∐_{i=1}^{k} Y_{i} → Z → 0 is an almost split sequence of modules in C, then k ≤ 2 and k = 2 with ℓ(Y_{1}) ≤ ℓ(Y_{2}) implies that ℓ(Y_{1}) < ℓ(X) < ℓ(Y_{2}). Examples of such components are the regular components over a hereditary artin algebra. An indecomposable module X belonging to a quasi-serial component is called quasi-serial, and X is quasi-simple if there exists no irreducible monomorphism ending in X. We shall use the following properties of a quasi-serial module.

Lemma 6 To each quasi-serial module X corresponds a unique family (X_{i})_{i∈ℕ} of indecomposables such that

(1) X_{1} is quasi-simple and X = X_{s} for some s ∈ ℕ;

(2) there exists an irreducible monomorphism ψ_{i}: X_{i} → X_{i+1} for all i ∈ ℕ.

Proof : See [8].

The number s ∈ ℕ with X = X_{s} is called the quasi-length of X and it is convenient to put X_{0} = 0.

Lemma 7 Let X be a quasi-serial module of quasi-length s and let 0 ≤ i ≤ s. Then there is an exact sequence 0 → X_{i} →^{ψ_{s−1}∘...∘ψ_{i}} X_{s} → (Tr D)^{i}(X_{s−i}) → 0. If 1 = i < s, then the induced morphism χ_{s}: X_{s} → Tr D(X_{s−1}) is irreducible.

Proof : Straightforward.

Each quasi-simple module determines uniquely a large indecomposable module in the following sense.

Theorem 8 Let X be a quasi-simple module. Then there exists a chain

X = X_{1} ⟶^{ψ_{1}} X_{2} ⟶^{ψ_{2}} …

of irreducible monomorphisms such that each X_{i} is indecomposable and X_{∞} = ⟶ lim X_{i} is a large denumerably generated indecomposable module. The module X_{∞} depends, up to isomorphism, only on X. If Y is a second quasi-simple module, then Y_{∞} ≅ X_{∞} if and only if Y ≅ X.

Proof : The existence of a chain X = X_{1} →^{ψ_{1}} X_{2} →^{ψ_{2}} … of irreducible monomorphisms follows from Lemma 6. For each i > 1 define φ_{i} = ψ_{i−1}∘ … ∘ψ_{1}. We obtain a family 0 → X →^{φ_{i}} X_{i} →^{χ_{i}} Tr D(X_{i−1}) → 0 of exact sequences where χ_{i} is irreducible for all i by Lemma 7. Thus the existence of a large denumerably generated indecomposable module X_{∞} = ⟶ lim X_{i} follows from Theorem 1 and it remains to show its uniqueness. First one observes that the X_{i}’s are uniquely determined by X and that any morphism ψ′_{i}: X_{i} → X_{i+1} satisfying φ_{i+1} = ψ′_{i}∘φ_{i} is irreducible. The latter guarantees that the module ⟶ lim X_{i} constructed in Theorem 1 is actually a limit where each morphism X_{i} → X_{i+1} is irreducible. Now assume an arbitrary choice of irreducible morphisms ψ′_{i}: X_{i} → X_{i+1}. It follows from [8, 4.1] that there exist isomorphisms μ_{i}: X_{i} → X_{i} satisfying μ_{i+1}∘ψ_{i} = ψ′_{i}∘μ_{i} for all i. Taking direct limits the morphism ⟶ lim μ_{i} gives an isomorphism between ⟶ lim X_{i} taken over the ψ_{i}’s and ⟶ lim X_{i} taken over the ψ′_{i}’s. Finally suppose that Y is a second quasi-simple such that Y_{∞} ≅ X_{∞}. The morphism φ = ⟶ lim φ_{i}: X → X_{∞} composed with the isomorphism X_{∞} → Y_{∞} factors through some Y_{j} since X is finitely generated. We denote the corresponding morphism by χ: X → Y_{j}. It is not hard to check that there exists an isomorphism ν: X_{j} → Y_{j} such that χ = ν∘φ_{j} and therefore X ≅ Y.

Let C be the center of the artin algebra Λ and denote by I an injective envelope of C/ rad(C). Using the functor D = Hom_{C}( , I) between Mod(Λ) and Mod(Λ^{op}) which induces a duality between mod(Λ) and mod(Λ^{op}) we obtain the following consequence of Theorem 8.

Corollary 9 Let X be a quasi-simple module. Then there exists a chain

… ⟶^{ψ^{2}} X^{2} ⟶^{ψ^{1}} X^{1} = X

of irreducible epimorphisms such that each X^{i} is indecomposable and X^{∞} = ⟵ lim X^{i} is a large module which depends, up to isomorphism, only on X.

Proof : The Λ^{op}-module Y = D(X) is quasi-simple and admits a chain Y = Y_{1} →^{φ_{1}} Y_{2} →^{φ_{2}} … of irreducible monomorphisms. Applying D we obtain a chain … →^{D(φ_{2})} D(Y_{2}) →^{D(φ_{1})} D(Y_{1}) ≅ X of irreducible epimorphisms with X^{∞} = ⟵ lim D(Y_{i}) ≅ D(⟶ lim Y_{i}) = D(Y_{∞}).

Remark 10 We will show in a subsequent paper that the module X_{∞} is Σ-pure-injective provided that (Tr D)^{n}(X) ≅ X for some n ∈ ℕ. Of course, the module X^{∞} is pure-injective since it is of the form D(Y) for some Λ^{op}-module Y. We do not know of any example where the module X^{∞} decomposes.

Let X be a quasi-serial module of quasi-length s and fix a chain X = X_{s} →^{ψ_{s}} X_{s+1} →^{ψ_{s+1}} … of irreducible monomorphisms such that X_{i} is indecomposable for all i > s. We denote by ψ_{X}: X → ⟶ lim X_{i} = X_{∞} the induced monomorphism. The module X_{∞} depends only on X according to Lemma 6 and Theorem 8. Moreover, the morphism ψ_{X} is unique in the sense that for any different choice of irreducible monomorphisms ψ′_{i}: X_{i} → X_{i+1}, i ≥ s, there exists an isomorphism μ: X_{∞} → X_{∞} with ψ′_{X} = μ∘ψ_{X}. Finally note that ψ_{X} induces an exact sequence

0 ⟶ X ⟶^{ψ_{X}} X_{∞} ⟶ (Tr D)^{s}(X)_{∞} ⟶ 0.

This follows from Lemma 7. Analogously, there is an induced epimorphism ψ^{X}: X^{∞} → X for X^{∞} = ⟵ lim X^{i} which is part of an exact sequence

0 ⟶ (D Tr)^{s}(X)^{∞} ⟶ X^{∞} ⟶^{ψ^{X}} X ⟶ 0.

We proceed with a discussion of the morphisms ψ_{X} and ψ^{X}. For each n ∈ ℕ and each pair of modules X, Y in mod(Λ) we denote by rad^{n}(X, Y) the morphisms in Hom_{Λ}(X, Y) which belong to the n-th power of the Jacobson radical of mod(Λ). As usual rad^{∞}(X, Y) = ∩_{n∈ℕ} rad^{n}(X, Y). Using the above notation we have the following lemma.

Lemma 11 Let X = X_{s} be a quasi-serial module and Y be indecomposable. If φ ∈ rad^{n}(X, Y) and n ≥ s, then there exists a morphism χ ∈ rad^{n−s}(X_{s+1}, Y) such that φ = χ∘ψ_{s}.

Proof : First suppose that n = s. For each i, 0 ≤ i ≤ s, there is an exact sequence 0 → X_{i} →^{ψ_{s−1}∘...∘ψ_{i}} X →^{π_{i}} Z_{i} → 0 with Z_{i} = (Tr D)^{i}(X_{s−i}) according to Lemma 7. Let i be maximal such that there are morphisms α: X_{s+1} → Y and β: Z_{i} → Y with φ = α∘ψ_{s} + β∘π_{i}. The assertion follows if i = s since π_{s} = 0. Therefore assume that i < s. Using the property of the almost split sequence starting in Z_{i} we deduce that β is an isomorphism. Thus φ ∉ rad^{i+1}(X, Y) since π_{i} ∉ rad^{i+1}(X, Z_{i}) by [7, Theorem 13.3] and Hom_{Λ}(X_{s+1}, Z_{i}) = rad^{i+1}(X_{s+1}, Z_{i}). This contradiction finishes the first part of our proof. Now suppose that n ≥ s. There are morphisms φ_{i}: X → Y_{i} and χ_{i} ∈ rad(Y_{i}, Y_{i+1}) for each i ∈ ℕ such that Im(Hom_{Λ}(φ_{i}, )) = rad^{i}(X, ) and φ_{i+1} = χ_{i}∘φ_{i} for all i ∈ ℕ [4, V, Lemma 7.10]. Using the first part we find a morphism χ_{0}: X_{s+1} → Y_{s} such that φ_{s} = χ_{0}∘ψ_{s}. Thus φ_{n} = χ_{n−1}∘ … ∘χ_{s}∘χ_{0}∘ψ_{s} with χ_{n−1}∘ … ∘χ_{s}∘χ_{0} ∈ rad^{n−s}(X_{s+1}, Y_{n}) and the proof is complete.

We are now in position to prove the following property of the morphism ψ_{X}: X → X_{∞}.

Theorem 12 Let φ: X → Y be a morphism between finitely generated indecomposable modules and suppose that X is quasi-serial.

(1) φ ∈ rad^{∞}(X, Y) if and only if there exists a morphism χ: X_{∞} → Y such that φ = χ∘ψ_{X}.

(2) If φ ∉ rad^{∞}(X, Y), then there exists a morphism χ: X_{∞} → Y_{∞} such that ψ_{Y} ∘φ = χ∘ψ_{X}.

Proof : (1) The inclusion Im(Hom_{Λ}(ψ_{X}, Y)) ⊆ rad^{∞}(X, Y) is obvious from the definition of ψ_{X}. Therefore let φ ∈ rad^{∞}(X, Y). We fix a chain X_{1} →^{ψ_{1}} X_{2} →^{ψ_{2}} … of irreducible monomorphisms with X_{1} quasi-simple and X = X_{s} for some s ∈ ℕ. It follows from the previous lemma that for all i ∈ ℕ any morphism χ_{i} ∈ rad^{∞}(X_{i}, Y) can be extended to a morphism χ_{i+1} ∈ rad^{∞}(X_{i+1}, Y) such that χ_{i} = χ_{i+1}∘ψ_{i}. Here one uses the fact that rad^{∞}(X_{i+1}, Y) = rad^{n}(X_{i+1}, Y) for some n ∈ ℕ. Taking χ_{s} = φ we obtain a morphism χ = ⟶ lim χ_{i} which has the desired property.

(2) Similar to (1).

Using again the functor D = Hom_{C}( , I) we obtain the analogous property of the mor-

phism ψ^{X}: X^{∞} → X.

Corollary 13 Let φ: Y → X be a morphism between finitely generated indecomposable mod-

ules and suppose that X is quasi-serial.

(1) φ ∈ rad^{∞}(Y, X) if and only if there exists a morphism χ: Y → X_{∞} such that φ =

ψ^{X} ∘χ.

(2) If φ ∉ rad^{∞}(Y, X), then there exists a morphism χ: Y^{∞} → X^{∞} such that φ∘ψ^{Y} =

ψ^{X} ∘χ.

References

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[4] M. Auslander, I. Reiten and S. Smalø, Representation theory of artin algebras, Cambridge University Press, 1995.

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[8] C.M. Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978) 235–255.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

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