Free Submodules of Indecomposable Modules

Claus Michael Ringel

ABSTRACT. Let Λ be an artin algebra which has only finitely many isomorphism classes of non-faithful indecomposable modules. We show that large indecomposable Λ-modules contain large free submodules.

1. Main Proposition.

Let Λ be an artin algebra. The modules to be considered will be finitely generated left Λ-modules.

Proposition. Let Λ be an artin algebra which has only finitely many isomorphism classes of non-faithful indecomposable modules. Then, for every natural number t, there is a bound b_{t} such that any indecomposable Λ-module of length at least b_{t} contains a free submodule of rank t.

We may formulate some consequences of the assertion:

If M is an indecomposable module of length at least b_{t}, and N is any module of length t, then N occurs as a subfactor of M (this means that there are submodules M′′ ⊆ M′ ⊆ M with M′/M′′ isomorphic to N).

Namely, if N is a module of length t, then N is a factor module of _{Λ}Λ^{t}, thus if the submodule M′ of M is free of rank t, there is M′′ ⊆ M′ such that M′/M′′ is isomorphic to N.

Assume that we know that there are infinitely many isomorphism classes of modules of length t. Then any indecomposable module of length at least b_{t} has infinitely many isomorphism classes of subfactors.

It is the last assertion which seems to be of interest, since it belongs to the focus of the Brauer-Thrall conjectures: to the problem of relating families of indecomposable modules of a fixed length to large indecomposable modules. We will provide further remarks on this relationship in section 3.

1991 Mathematics Subject Classification. Primary 16G60. Secondary 16D40, 16D70, 16G20.

C. M. Ringel

2. Proof.

The proof of Proposition will be given in this section. We need some preparations. The length of a module M will be denoted by |M|, let d = |_{Λ}Λ|.

We say that a module M has height at least h ≥ 1 provided there is a filtration

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{h} = M

such that all the subfactors M_{i}/M_{i−1} (with 1 ≤ i ≤ h) are faithful (and M has height 0, provided M is not faithful). The height h(M) of M is by definition the maximum of all numbers h such that M has height at least h; note that the height h(M) is obviously bounded by the length |M|.

Lemma 1. Assume that h(M) ≥ d. Then _{Λ}Λ can be embedded into M.

Proof: We show by induction on r the following: if the height of M is at least r, and r ≤ d, then there is a map u: _{Λ}Λ → M with image of length at least r. The induction starts with r = 0 and u the zero map. Assume that the assertion has been shown for some r < d. Consider a module M of height r+1, say with a filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{r+1} such that all the factors M_{i}/M_{i−1}, for 1 ≤ i ≤ r + 1, are faithful. Now M/M_{1} has height at least r, thus there is a map u: _{Λ}Λ → M/M_{1} with image I of length at least r. Since _{Λ}Λ is a free module, there exists a lifting u′: _{Λ}Λ → M such that πu′ = u, where π: M → M/M_{1} is the canonical epimorphism with kernel M_{1}. Consider the image I′ of u′. Case 1: I′ ∩ M_{1} ≠ 0. Then |I′| > |I|, thus |I′| ≥ r + 1. Case 2: I′ ∩ M_{1} = 0, thus |I′| = |I| ≥ r. If |I| > r, then we are done, thus assume |I| = r. Let u′(1) = x, thus u′(λ) = λx. Since |I| = r < d = |_{Λ}Λ|, we see that there is some left ideal L of Λ such that L ≠ 0, but Lx = 0. Take a non-zero element λ_{0} ∈ L. Since M_{1} is faithful, there is y ∈ M_{1} with λ_{0}y ≠ 0, thus λ_{0} is not contained in the kernel of the map v: _{Λ}Λ → M_{1} defined by v(1) = y. Now, consider the map u′′ = [ \begin{matrix}u′ \\ v \end{matrix} ] : _{Λ}Λ → I ⊕ M_{1} ⊆ M. The kernel L′′ of u′′ is the intersection of the kernel L of u′ and the kernel of v. Since the kernel of v does not contain λ_{0} (and λ_{0} ∈ L), we see that L′′ is a proper subset of L, thus the image of u′′ has length greater than r. This completes the proof.

Corollary. Assume that h(M) ≥ dt. Then the free Λ-module of rank t can be embedded into M.

Proof: Let 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{t} = M be a filtration, where any factor M_{i}/M_{i−1} for 1 ≤ i ≤ t has height at least d. According to the previous lemma, there is an embedding u_{i}: _{Λ}Λ → M_{i}/M_{i−1}, and, since _{Λ}Λ is a free module, we can lift the embedding to an embedding u′_{i}: _{Λ}Λ → M_{i}. Let I_{i} be the image of u′_{i}. Since the composite map _{Λ}Λ → M_{i} → M_{i}/M_{i−1} still is injective, we see that I_{i} ∩ M_{i−1} = 0. But this means that ∑_{i=1}^{t} I_{i} actually is a direct sum, thus isomorphic to _{Λ}Λ^{t}.

Let k be the center of Λ, it is a commutative artinian ring (and Λ is a finitely generated k-module). For any pair of Λ-modules M, N, the group Ext^{1}(M, N) is a k-module of finite length, where k is the center of Λ, and we write dim Ext^{1}(M, N) in order to denote its length as a k-module.

Ext-Lemma. Let N be an indecomposable module. Let

0 ⟶ N^{n} ⊕ N′ ⟶^{u} X ⟶ Q ⟶ 0

Free Submodules

be an exact sequence of Λ-modules. If dim Ext^{1}(Q, N) < n, then for at least one of the canonical inclusions u_{i}: N → N^{n} ⊕N′, the composition uu_{i} is a split monomorphism.

This is a basic and well-known result in homological algebra, it is one of the ingredients used by Roiter in [Ro], see also [Ri].

We assume from now on that Λ is an artin algebra which has only finitely many isomorphism classes of non-faithful indecomposable modules, let N_{1}, …, N_{m} be such modules, one from each isomorphism class. Let

e_{i} = max{dim Ext^{1}(S, N_{i}) | S a simple Λ-module}, and
e′_{i} = max{dim Ext^{1}(N_{i}, S) | S a simple Λ-module}.

We can use these numbers in order to bound other Ext-groups. For example, for every module M we see by induction on the length of M that

dim Ext^{1}(N_{i}, M) ≤ |M|e′_{i}.

Also, let us introduce the following numbers:

c = 1 + ∑ e_{i}|N_{i}| and c′ = 1 + ∑ e′_{i}|N_{i}|.

Call M f-minimal provided M is indecomposable and faithful, and no proper indecomposable submodule is faithful.

Lemma 2. An f-minimal module M has length at most c.

Proof: Let M a be f-minimal. Let M′ be a maximal submodule of M. Our assumption implies that any indecomposable direct summands of M′ is of the form N_{i}, thus M′ = ⨁_{i=1}^{m} N_{i}^{x_{i}}. According to the Ext-Lemma, x_{i} ≤ e_{i}, since otherwise M would have a direct summand isomorphic to N_{i}, but M is indecomposable and faithful. This shows that |M| = |M/M′| + |M′| = 1 + ∑ x_{i}|N_{i}| ≤ 1 + e_{i}|N_{i}| = c.

Lemma 3. Any module M of height h(M) = t has a filtration

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{s} ⊆ M

with s ≤ t such that the factors M_{i}/M_{i−1} for 1 ≤ i ≤ s are f-minimal and M/M_{s} is a direct sum of non-faithful modules.

Proof: We are going to define M_{i} inductively, as follows: Let M_{0} = 0. Assume that we have constructed already 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{r} such that all the factors M_{i}/M_{i−1} are f-minimal, for 1 ≤ i ≤ r. If M/M_{r} is a direct sum of nonfaithful modules, let s = r. Thus assume M/M_{r} has an indecomposable direct summand which is faithful, say U. Then there is a submodule U′ of U which is f-minimal. Write U′ = M_{r+1}/M_{r}, where M_{r} ⊂ M_{r+1} ⊆ M. This procedure will stop eventually, thus we obtain the filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{s} ⊆ M where M_{i}/M_{i−1} for 1 ≤ i ≤ s are f-minimal and M/M_{s} is a direct sum of non-faithful modules. Since there are s factors which are faithful, we see that s ≤ h(M) = t.

Lemma 4. Assume that M is indecomposable and faithful. Then

|M| ≤ cc′h(M).

Proof. According to Lemma 3, there is a filtration

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{s} ⊆ M

with s ≤ t such that the factors M_{i}/M_{i−1} for 1 ≤ i ≤ s are f-minimal and M/M_{s} is a direct sum of non-faithful modules. By Lemma 2, |M_{i}/M_{i−1}| ≤ c, for 1 ≤ i ≤ s, thus |M_{s}| ≤ cs.

Write M/M_{s} = ⨁ N_{i}^{x_{i}}. According to the dual of the Ext-Lemma, we have

x_{i} ≤ dim Ext^{1}(N_{i}, M_{s}) ≤ |M_{s}|e′_{i} ≤ cse′_{i}

Thus

|M| = |M_{s}| + |M/M_{s}| ≤ cs + ∑ x_{i}|N_{i}| ≤ cs + ∑ cse′_{i}|N_{i}|
= cs(1 + ∑ e′_{i}|N_{i}|) = cc′s ≤ cc′t.

This completes the proof.

Proof of Proposition. Let b_{t} = cc′dt. If |M| ≥ b_{t}, then

h(M) ≥ \frac{1}{cc′} |M| ≥ \frac{1}{cc′} b_{t} = dt,

where the first inequality sign comes from Lemma 4. Corollary of Lemma 1 implies that _{Λ}Λ^{t} can be embedded into M, as we wanted to show.

3. Remarks concerning the second Brauer-Thrall conjecture.

In his famous solution of the first Brauer-Thrall conjecture [Ro], Roiter has shown how infinite families of pairwise non-isomorphic modules with fixed length can be used in order to construct large indecomposable modules (see also [Ri]). Of course, infinite families of pairwise non-isomorphic modules with fixed length can only exist in case Λ is a ring with infinitely many elements (or, equivalently, in case Λ is a module-finite k-algebra, where k is a commutative ring with infinitely many elements). Under this assumption, one may conversely ask whether it is possible to construct infinite families of pairwise non-isomorphic modules with fixed length, starting from large indecomposable modules. The second Brauer-Thrall conjecture (known to hold for finite-dimensional k-algebras where k is an infinite perfect field) asserts that any representation infinite k-algebra has infinite families of pairwise non-isomorphic modules with fixed length, however it does not predict how to get hold of such a family. We propose the following conjecture:

Conjecture. Assume that Λ is an infinite artin algebra which is of infinite representation type. Then there exists a bound b such that any indecomposable module of length greater than b contains infinitely many isomorphism classes of subfactors.

Let us observe that for some well-known algebras, such a bound can easily be calculated (and it turns out that b often may be chosen to be amazingly small). For example, for the Kronecker quiver take b = 2: Any indecomposable Kronecker module of length greater than 2 contains infinitely many isomorphism classes of indecomposable subfactors. Namely, any indecomposable Kronecker module M of length at least 3 contains an indecomposable module of length 3 either as a submodule (in case M is preprojective or regular) or else as a factor module (in case M is preinjective or regular). There are two isomorphism classes of indecomposable modules N of length 3: the module N may be either projective or else injective. The indecomposable projective module of length 3 has infinitely many isomorphism classes of factor modules, the indecomposable injective module of length 3 has infinitely many isomorphism classes of submodules. Actually, the same bound b = 2 works even for all the generalized Kronecker algebras (these are the path algebras of the quivers with two vertices a, b and a finite number of arrows a → b).

In case Λ is an arbitrary tame hereditary algebra, the conjecture can be shown to hold; the bounds b can be calculated in all these cases, they are larger than in the Kronecker case, but still quite small.

If Λ is a string algebra, then b = 2sp works, where s = s(Λ) is the number of isomorphism classes of simple Λ-modules and p = p(Λ) the maximal length of an indecomposable projective Λ-module. For a string algebra Λ, the indecomposable Λ-modules are well-known and easy to describe, see for example [BR] for the terminology used here. Consider a string module M of length greater than b. Its socle has length at least 2s + 1, thus one of the simple modules occurs in the socle with multiplicity at least 3. The module M = M(w) is described by some word w, and there are the following three possibilities: first, w contains (at least) twice a subword of the form β^{−1}α, second, w starts with α and contains β^{−1}α as a subword, or third, w ends with β^{−1} and contains β^{−1}α as a subword. Always, we obtain all the primitive band modules M(w′, λ) a factor modules of M, where w′ is a primitive cyclic word starting with α and ending with β^{−1}. A similar argument can be used in case we consider a band module of length greater than b.

Let us return to the second Brauer-Thrall conjecture, thus we assume that we consider an artin algebra Λ with infinitely many elements. Recall that there is one class of algebras which can be treated quite easily: those with an infinite set of two-sided ideals. As Jans [J] has shown in this situation (at least if Λ is a finite dimensional algebra over a field k) there exists a one-parameter family of isomorphism classes of indecomposable modules which are local and say of length q (they are annihilated by pairwise different ideals, in particular, none of these modules is faithful); and one can form self-extensions of these modules in order to obtain corresponding one-parameter families of indecomposable modules of length nq, for all n ∈ ℕ_{1}. Thus, the interesting class of algebras which have to be considered are those with only finitely many two-sided ideals.

An artin algebra which is not of finite representation type will be said to be representation-infinite. In case Λ is representation-infinite, but any proper factor algebra is of finite representation type, then Λ will be said to be minimal representation-infinite.

Lemma. Let Λ be an artin algebra with infinitely many elements. The following properties are equivalent:

(i) Λ is minimal representation-infinite and has only finitely many two-sided ideals. (ii) Λ is representation-infinite and has only finitely many isomorphism classes of

non-faithful indecomposable modules.

Proof: (i) ⟹ (ii). If there are only finitely many two-sided ideals, and if for every non-zero two-sided ideal I, the factor algebra Λ/I is of finite representation type, then there are only finitely many isomorphism classes of non-faithful indecomposable modules.

(ii) ⟹ (i). Note that condition (ii) immediately implies that Λ is minimal representation-infinite. But (ii) also implies that there are only finitely many ideals, as Jans [J] has shown.

References.

[BR] M.C.R.Butler, C.M.Ringel: Auslander-Reiten sequences with few middle terms,

with applications to string algebras. Comm. Algebra 15 (1987), 145-179.

[J] J.P.Jans: On the indecomposable representations of algebras. Ann. of Math. 66 (1957), 418-429.

[Ri] C.M.Ringel: The Gabriel-Roiter measure. (Preprint)

[Ro] A.V.Roiter: Unboundedness of the dimension of the indecomposable representations of an algebra which has infinitely many indecomposable representations. Izv. Akad. Nauk SSSR. Ser. Mat. 32 (1968), 1275-1282.

Fakultät für Mathematik, Universität Bielefeld, POBox 100 131, D-33 501 Bielefeld

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