TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 5, May 1998, Pages 2009–2031 S 0002-9947(98)01929-1

COMPOSITION FACTORS OF INDECOMPOSABLE MODULES

MARIA IZABEL RAMALHO MARTINS

Abstract. Let Λ be a connected, basic finite dimensional algebra over an algebraically closed field. Our main aim is to prove that if Λ is biserial, its ordinary quiver has no loop and every indecomposable Λ-module is uniquely determined by its composition factors, then each indecomposable Λ-module is multiplicity-free.

In this article a k-algebra Λ means a finite dimensional k-algebra, where k is a fixed algebraically closed field. Denote by Λ-mod the category of all finitely generated left Λ-modules and by Λ-ind the full subcategory of Λ-mod defined by one representative of each isomorphism class of indecomposable Λ-modules. All modules and maps are in Λ-mod. A module M is called multiplicity-free if its composition factors are pairwise non-isomorphic.

We say that Λ-ind is determined by composition factors (multiplicity-free) if every module in Λ-ind is uniquely determined by its composition factors (respectively, multiplicity-free). For example, it is well-known that an algebra Λ is semisimple if and only if each of its modules is uniquely determined by composition factors. Though the composition factors do not determine arbitrary modules over nonsemisimple algebras, it is an interesting question to know under which conditions it has the property that Λ-ind is determined by composition factors. In [5], Auslander and Reiten obtained sufficient conditions for Λ-ind to be determined by composition factors. Besides this question it is also an interesting question to know when an algebra has the property that Λ-ind is multiplicity free, and so find out if there is any relationship between those properties. It is well-known, for example, that if Λ is a representation-finite hereditary k-algebra then Λ-ind is determined by composition factors (see [2]), although it may not hold true in general that “Λ-ind is multiplicity-free”. It is not difficult to show that, among the representation-finite hereditary k-algebras, the ones of the form Λ = kQ, with Λ-ind multiplicity-free, coincide with those whose quiver Q is of type A_{n}.

According to [12], if Λ is a k-algebra such that Λ-ind is multiplicity-free, then Λ is a representation-finite biserial and Schurian algebra. Recall that an algebra Λ is biserial if the radical of any indecomposable non-uniserial projective, left or right, Λ-module is a sum of two uniserial submodules whose intersection is simple or zero. Well-known examples of biserial algebras are Nakayama algebras and iterated tilted algebras of type A_{n} (see [1] for the definition).

With respect to a relationship between the properties Λ-ind being determined by composition factors and Λ-ind being multiplicity-free, the results of Pogorzaly and Skowroński (see [12], th.1) together with results of Skowroński and Waschbüsch (see

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors September 19, 1995 and, in revised form, August 1, 1996.

1991 Mathematics Subject Classification. Primary 16G20, 16G60.

[14], cor. of th.1) show that the conditions (i) Λ is biserial and Λ-ind is determined by composition factors and (ii) Λ-ind is multiplicity-free are equivalent, if Λ is assumed to be a factor of a hereditary algebra. (Recall that a k-algebra Λ is a factor of a hereditary algebra if and only if the corresponding ordinary quiver has neither loops nor oriented cycles.)

This leads us to the following questions: for which k-algebras does the implication (i) ⇒ (ii) or (ii) ⇒ (i) hold? Can these hold even if Λ is not a factor of a hereditary algebra? With respect to the implication (ii) implies (i) the answer is not always. We shall exhibit, in section 3, an example where Λ is a biserial k-algebra, not Nakayama, such that (ii) is satisfied but (i) is not. (It is very simple to find a Nakayama algebra with such properties).

The answer to the implication (i) implies (ii) is the main aim of this article, and it is given by the following theorem.

Theorem 6.2. Let Λ be a connected, basic finite dimensional algebra over an algebraically closed field, whose ordinary quiver has no loop. Assume that Λ-ind is determined by composition factors. Then Λ-ind is multiplicity-free.

In order to formulate the main result we used the diagramatic structure of the representation-finite biserial k-algebras Λ given by Skowroński and Pogorzaly (in [12], th.2). We carry this structure over to a certain “local” structure of some indecomposable modules, in the case that the ordinary quiver Q of Λ has no loop and all oriented cycles in kQ are in the relation set; and this allows us to prove an interesting result which is useful for the proof of theorem 6.2 , which is the following theorem.

Theorem 6.1. Let Λ = k(Q_{Λ}, R_{Λ}) be a representation-finite biserial bound quiver algebra. Suppose that the quiver Q_{Λ} has no loop and that all oriented cycles of kQ_{Λ} are in R_{Λ}. If M is a multiplicity-free indecomposable Λ-module, then either M is projective-injective non-uniserial or M is of type A_{n}.

As a consequence of theorems 6.1 and 6.2 we have that if Λ is a biserial kalgebra whose ordinary quiver has no loop, and such that Λ-ind is determined by composition factors, then there are two types of indecomposable modules: the projective-injective non-uniserial modules and the modules of type A_{n}.

The results in this article are part of my Ph.D. thesis, written under Professor Hector A. Merklen at University of São Paulo. I thank Professor Merklen for very stimulating discussions and for his useful remarks during my work.

1. Preliminaries

Given a k-algebra Λ and a module M, we denote by K_{0}(Λ) the Grothendieck group of Λ and by [M] the image of M in K_{0}(Λ). We will identify a k-algebra Λ with its associated bound quiver algebra k(Q_{Λ}, R_{Λ}) (see [8], [9]), where Q_{Λ} denotes the ordinary quiver of Λ. Consequently we will identify the category Λmod with the category (Q_{Λ}, R_{Λ})-mod of the finite k-representations of (Q_{Λ}, R_{Λ}) ([8], [9]). This enables us to denote by P_{a} (I_{a}) the indecomposable projective (injective) module which corresponds to the vertex a of Q_{Λ}, and by S_{a} the simple module associated to the vertex a. If (Q, R) is a quiver with relations, we denote by Q^{op} the quiver obtained from Q by reversing arrows and by γ^{∗} the arrows of Q^{op} to which correspond the arrows γ of Q. So (Q^{op}, R^{op}) is the opposite quiver with relations of (Q, R), where R^{op} = {ρ^{∗} ∈ kQ^{op} : ρ ∈ R}. To each module M we associate a subquiver of Q_{Λ}, which is denoted by Q_{M}, as follows. The set of vertices of Q_{M} is the set of the vertices x of Q_{Λ} for which M(x) ≠ (0), and its set of arrows is formed by the arrows α of Q_{Λ} for which M(α) ≠ 0. Sometimes we refer to the set of vertices of the quiver of M as the support of M and denote it by supp M. It is easy to verify that if M is indecomposable, then Q_{M} is connected. We say that a module M is of type A_{n} if the underlying graph of the quiver Q_{M} is a diagram A_{n} and if M(x) ≅ k, for each x ∈ supp M, and the k-morphism M(α) is a scalar multiple of the identity, for each arrow α of Q_{M}. For an algebra Λ and any Λ-module M we will denote by I_{M} the Λ-injective envelope of M, by topM the top of M, by socM the socle of M, by radM the radical of M, and by l(M) the length of M. Finally, by D we denote the duality Hom_{k}(−, k), by τ we denote the Auslander-Reiten operator which maps the non-projective indecomposables one-to-one to the non-injective indecomposables (it has a natural inverse τ^{−1}), and by Γ_{Λ} the Auslander-Reiten quiver of Λ (see [3], [4], or [9]).

Remark 1.1. Let Λ be a k-algebra such that Λ-ind is determined by composition factors. Then Λ is representation-finite. This is an immediate consequence of BT II (see [7], [11] or [13]). It is a nice problem to prove this without using BT II. It follows from this that the ordinary quiver of Λ has neither multiple arrows nor diagrams Ã_{n} without relations, where n≥3.

2. Nakayama algebras

In this section we consider our questions in the easy case of Nakayama algebras. Recall that for a Nakayama algebra Λ, by definition, all indecomposable projective and injective Λ-modules are uniserial. In particular, in this section the algebras mentioned are Artin algebras (that is, an artinian ring Λ with 1 such that its centre contains an artinian subring over which Λ is finitely generated as a module). We refer to [2], [6] and [8] for well known facts that we shall use here freely. Of course, as Λ is assumed connected, we assume that all simple Λ-modules lie in one τ-orbit. Thus, we can also order the representatives P_{1}, P_{2}, ..., P_{n} of all indecomposable projective modules according to the Kupisch series (see [6]), that is, in such a way that top P_{i+1} ≅ τ^{−1}(top P_{i}) for i = 1, 2, ..., n − 1 and, if l(P_{1}) ≠ 1, top P_{1} ≅ τ^{−1}(top P_{n}).

Proposition 2.1. Let Λ be a connected, basic Nakayama algebra and n be the rank of the Grothendieck group of Λ. If n = 1 or n ≥ 2 and there is a simple projective Λ-module, then Λ-ind is always determined by composition factors. If, on the contrary, n ≥ 2 and l(P) ≥ 2 for all indecomposable projective Λ-module, then Λ-ind is determined by composition factors if and only if l(P) ≤ n, for all indecomposable projective P, and there is at most one of them of length n.

Proof. The first of these assertions is an easy consequence of well known properties related to the Kupisch series of Λ, and the same is the case for the “only if” part of the last assertion.

Let us assume now that l(M)≤n, for all M in Λ-ind. If M is not projective, then l(M)<n and, since it is determined by its top and length, M is uniquely determined by its composition factors. But this is clearly true for any indecomposable projective Λ-module P with l(P)<n. Hence the result follows.

When Λ is a Nakayama algebra, we also establish necessary and sufficient conditions for Λ-ind to be multiplicity-free.

Proposition 2.2. Let Λ be a connected, basic Nakayama algebra and n be the rank of the Grothendieck group of Λ. Then Λ-ind is multiplicity-free if and only if every projective indecomposable Λ-module has length at most n.

Proof. This is an immediate consequence of properties related to the Kupisch series and the fact that every indecomposable Λ-module is local.

3. Some conditions for Λ-ind to be determined by composition factors

In this section we will establish some necessary conditions for Λ-ind to be determined by composition factors, in the case that Λ is a bound quiver algebra.

Let Λ = k(Q, R) be a bound quiver algebra, where k is a fixed field. Ringel, in [13], describes some algorithms that give a way to control the category Λ-mod when, for example, we delete certain vertices or arrows of Q. It allows us to analyse Λ-mod by simpler quivers whose associated bound quiver algebra may have some suitable properties. We will use two of those algorithms, namely AL.1: deleting for vertices and AL.2: deleting for arrows (see [13]). These processes allow us to obtain a simpler quiver with relations (Q′, R′), such that (Q′, R′)-mod is a full subcategory of (Q, R)-mod. Furthermore, they can be applied repeatedly and alternately any given number of times.

Proposition 3.1. Let Λ be a connected, basic k-algebra. If Λ-ind is determined by composition factors, then the quiver with relations (Q_{Λ}, R_{Λ}) such that Λ = k(Q_{Λ}, R_{Λ}) satisfies the following conditions:

(i) Q_{Λ} has no multiple arrow or subquivers Ã_{n} without relations, where n≥3;

(ii) Q_{Λ} has no subquiver Q′: [figure: β; α; b; a] (or [figure: ^{β′}; ^{α′}; b; a]), with βα ∉ R_{Λ} (or α′β′ ∉R_{Λ}); (iii) kQ_{Λ} has no oriented cycle μ=α_{r}α_{r−1}...α_{2}α_{1}, with r ≥2, α_{i} ≠α_{j} if i≠j, such that μ∉R_{Λ} and every subpath of μ is not a summand of a generator of R_{Λ}.

Proof. Since Λ-ind is determined by composition factors, by remark 1.1, it follows that (i) is satisfied.

(ii) Suppose that Q_{Λ} contains a subquiver Q′. We can reduce to the case that Q_{Λ} = Q′. In fact, if Q′ is a proper subquiver of Q_{Λ}, deleting the vertices x ∉ {a, b} and deleting the arrows γ ∉{α, β} (or γ ∉{α′, β′}), applying AL.1 and AL.2 repeatedly and alternately, we obtain the algebra Λ′ = k(Q′, R′), where βα ∉ R′ (or α′β′ ∉ R′), whose (Q′, R′)-mod is a full subcategory of Λ-mod. We will write the proof of the first case only, since the other one follows similarly.

Since R_{Λ} is an admissible ideal of kQ_{Λ} and βα ∉ R_{Λ}, there are r and s in ℤ greater than or equal to 2 such that α^{r} and βα^{s} are in R_{Λ}. Without loss of generality we can suppose that r = s = 2. Let us consider the indecomposable projective P_{a} = Λ(e_{a} + R), where e_{a} is the trivial path of kQ_{Λ} with origin at a. Then rad P_{a} =Λ(α+R)⊕Λ(β+R) with Λ(β+R)=k(β+R) ≅ S_{b}. Therefore [P_{a}]= 2 [S_{a}] + 2 [S_{b}] in K_{0}(Λ), and from this it follows that the module M = P_{a}/rad^{2}P_{a} is indecomposable and such that [M] = 2 [S_{a}] + [S_{b}] and soc M = S_{a} ⊕ S_{b}. Let us consider the indecomposable injective I_{b} = DP_{b}^{op}. Since Λ^{op} = k(Q^{op}, R^{op}), it is easy to see that [I_{b}] = [DP_{b}^{op}] = 2 [S_{a}] + [S_{b}] in K_{0}(Λ^{op}) and soc I_{b} = S_{b}. Hence we obtain two indecomposable modules, which are the modules M and I_{b}, having the same composition factors, but not isomorphic, which contradicts the assumption.

(iii) Suppose that there exists in kQ_{Λ} an oriented cycle μ as in (iii). Denote by Q′′ the subquiver of Q_{Λ} which is the cycle μ. Applying AL.1 and AL.2 to delete conveniently certain vertices and arrows, we obtain an algebra Λ′ = k(Q′′, R′′) such that u ∉ R′′ and whose (Q′′, R′′)-mod is a full subcategory of Λ-mod. But Λ′ = k(Q′′, R′′) is a connected Nakayama algebra such that the rank of K_{0}(Λ′) is r ≥2. Since μ∉R′′, it follows that there is an indecomposable projective Λ′-module P with l(P) > r (for example, P = P_{o(α_{1})}). So, by proposition 2.1, Λ′-ind is not determined by composition factors, and as a consequence Λ-ind is not determined by composition factors, which is a contradiction to the assumption on Λ-ind.

To conclude this section, we present an example of an algebra Λ whose Λ-ind is multiplicity-free but is not determined by composition factors, which shows that the property (ii) does not imply (i).

Example 3.1. Let Λ = k(Q, R), where Q is the quiver below and R is given by γβα=0 and αγ =0.

[figure: 3; 1; 2; 4; γ; δ; α; β]

Λ is a representation-finite biserial algebra, where dim_{k}Λ = 12, such that all indecomposable projective modules, except P_{1}, and all indecomposable injective modules are uniserial (P_{1} is not uniserial, since rad P_{1} = P_{4} ⊕ N, where N ≅ P_{2}/rad^{2} P_{2}). The Auslander-Reiten quiver of Λ, illustrated below, shows that Λind is multiplicity-free. But Λ-ind is not determined by composition factors, since, for example, P_{1} and P_{2} are nonisomorphic modules having the same composition factors.

Γ_{Λ} [figure: ^{0 01 1}; ^{1 01 1}; ^{P_{3}^{≅}}; ^{0 01 0}; ^{1 01 0}; ^{1 11 0}; ^{I_{1} ^{≅}}; ^{1 00 0}; ^{1 10 0}; ^{N^{≅}}; ^{1 11 1}; ^{P_{1}^{≅}}; ^{0 10 0}; ^{0 11 1}; ^{0 11 0}; ^{I_{2} ^{≅}}; ^{0 01 1}; ^{0 01 0}; ^{1 11 1}; ^{P_{2}^{≅}}; ^{≅^{I_{4}}}; ^{1 11 0}; ^{I_{3} ^{≅}}; ^{0 00 1}; ^{P_{4}^{≅}}]

4. Indecomposable extensions of a simple module

In [12], Pogorzaly and Skowroński settled the diagrammatic structure of the representation-finite biserial algebras, through the following theorem.

Theorem 4.1. (see [12], th.2) Any representation-finite biserial k-algebra Λ is isomorphic to a bound quiver algebra k(Q_{Λ}, R_{Λ}), where (Q_{Λ}, R_{Λ}) satisfies the following conditions, denoted by (SP):

(1) The number of arrows starting (ending) at any fixed vertex of Q_{Λ} is at most two.

(2) For any arrow α of Q_{Λ} there is at most one arrow β and at most one arrow γ such that βα and αγ are not in R_{Λ}.

(3) There is an upper bound for the length of paths in Q_{Λ} which are not in R_{Λ}.

(4) R_{Λ} is generated by paths and by the differences of pairs of parallel paths of Q_{Λ}.

From now on, when we refer to a representation-finite biserial k-algebra Λ, (Q_{Λ}, R_{Λ}) will denote a fixed bound quiver satisfying the conditions (SP) of the above theorem.

Remark 4.1. In a bound quiver (Q, R) which satisfies the above conditions (SP) it is easy to see that the following conditions are verified. (4.1.a) A pair of parallel paths of kQ, which are not in R, does not contain common arrows, and their vertices, except the origin and the end, are vertices in just one of these paths. (4.1.b) Given a path μ∈kQ−R, no proper subpath of μ is a summand of a generator of R. (4.1.c) Given an arrow α: i → j of Q, then the conditions (SP)1, 2, 3 determine the unique path not in R starting in α, which is the maximal path among all paths μ_{α} which are not in R and starting in α. We denote this path by w_{α}. Moreover, if Q does not contain loops or subpaths which are oriented cycles without repeated arrows, then the arrows of w_{α} are all distinct and each vertex of w_{α}, except the origin and the end, belongs to exactly two of these arrows.

To simplify the reading, we introduce some notations. Given a path w of Q, we denote by w_{1} the set of the arrows in w and by w_{0} the set of the vertices in w. The expression “a path u_{α}” means a path u starting in the arrow α. Moreover, we denote by û_{α} the oriented cycle of minimal length among all the oriented cycles (if they exist), starting in α, which are not in R.

Using these notations, we state (without proof) the following technical lemma.

Lemma 4.2. Let k(Q, R) be a bound quiver algebra, where (Q, R) satisfies the conditions (SP) of theorem 4.1 and Q contains neither loops nor double arrows. Let i be a fixed vertex of Q.

(a) Suppose that i is the origin of exactly one arrow α : i → j (notation i=o(α)). (a.1) If the end of w_{α} (notation e(w_{α})) is i, then w_{α} =(û_{α})^{r}, for some r ≥1. (a.2) If e(w_{α}) ≠ i, then either (i) w_{α} = u′_{α}(û_{α})^{r}, for some r ≥ 1, where u′_{α}

is a proper subpath of û_{α}, or (ii). w_{α} does not contain oriented cycles starting at i;

(b) Suppose that i is the origin of two distinct arrows α : i → j_{1} e β : i → j_{2}.

(b.1) If there are two oriented cycles, one starting in α and the other starting in β, then w_{α} = û_{α} and w_{β} = û_{β}, and such oriented cycles are parallel

paths (so w_{α} − w_{β} ∈R).

(b.2) If e(w_{α})=i and e(w_{β})≠i, then w_{α} = (û_{α})^{r}, for some r ≥1, w_{β} does not contain oriented cycles starting at i, (w_{α})_{0} ∩(w_{β})_{0} =i and (w_{α})_{1} ∩(w_{β})_{1} =

∅.

(b.3) If e(w_{α}) = e(w_{β}) ≠ i, then either (i) one of them, say w_{α}, contains an oriented cycle starting in α, and, in this case, w_{α} = w_{β}û_{α} and û_{α} does not contain the arrow β, or (ii) w_{α} and w_{β} are parallel (so, w_{α} − w_{β} ∈R).

(b.4) If e(w_{α}) ≠ e(w_{β}), with i ∈ {e(w_{α}), e(w_{β})}, then either (i) w_{α} = u′_{α}(û_{α})^{r}, for some r ≥ 1, where u′_{α} is a proper subpath of û_{α}, w_{β} contains no oriented cycle starting in β, (w_{α})_{0} ∩ (w_{β})_{0} = i and (w_{α})_{1} ∩ (w_{β})_{1} = ∅, or (ii) w_{α} = u_{β}û_{α} and w_{β} = γu_{β}, where u_{β} is a proper subpath of w_{β} starting at e(w_{α}), γ is a subpath of w_{β} starting at e(w_{α}) and such that (û_{α})_{0} ∩ (w_{β})_{0} = i, and (û_{α})_{1} ∩ (w_{β})_{1} = ∅ or (iii) w_{α} and w_{β} contain no oriented cycle starting at i, and i is the unique common vertex among them and they do not have common arrows.

The structure given by theorem 4.1, and the description in lemma 4.2 and proposition 3.1, allow us initially to prove the following theorem, where we shall use the notations above.

Theorem 4.3. Let Λ be a biserial k-algebra whose ordinary quiver Q_{Λ} does not contain any loop. Suppose that Λ-ind is determined by composition factors. Then each projective indecomposable Λ-module and each injective indecomposable Λmodule is multiplicity-free.

Proof. Using the duality D, it is enough to show that the indecomposable projective modules are multiplicity-free. By remark 1.1, Λ is representation-finite and, being a biserial algebra, by theorem 4.1 it is isomorphic to k(Q_{Λ}, R_{Λ}), where (Q_{Λ}, R_{Λ}) satisfies the conditions (SP) and the conditions of proposition 3.1.

Suppose that there is a projective indecomposable module P and a simple module S such that S is a composition factor of P with multiplicity greater than or equal to 2. So, P ≅ P_{i} for some vertex i of Q_{Λ}, S ≅ S_{j} for some vertex j of Q_{Λ} and dim_{k}P (j) ≥ 2. 1\underline{st} case: S ≅ S_{i} (i.e, i = j). Then there exists an oriented cycle in kQ_{Λ} − R_{Λ}, starting at i. Depending on whether i is the origin of one arrow or of two distinct arrows, we have one of the cases (a.1), (a.2.i), (b.1), (b.2), (b.3.i), (b.4.i) or (b.4.ii) in lemma 4.2. If we have case (b.1), then the maximal paths w_{α} and w_{β} are parallel and w_{α} − w_{β} ∈R_{Λ}. Hence,

rad P_{i} =Λ(α + R_{Λ}) + Λ(β + R_{Λ}) and soc P_{i} =k(w_{α} + R_{Λ})=k(w_{β} + R_{Λ}) ≅ S_{i}.

So the indecomposable modules M = P_{i}/Λ(β + R_{Λ}) and N = Λ(α + R_{Λ}) have the same composition factors (the simple modules S_{x}, where x ∈ (w_{α})_{0}), but they are not isomorphic, since top M ≅ S_{i} and top N ≅ S_{e(α)}. This is a contradiction to Λind being determined by composition factors. In any of the cases (a.1), (a.2.i), (b.2), (b.3.i), (b.4.i) or (b.4.ii), using the notation of the last lemma, we can consider the oriented cycle û_{α}. By the minimality of the length of û_{α}, it contains no repeated arrows and, by remark 4.1.b, this cycle contains no subpath which is a summand of a generator of R_{Λ}. Then we get a contradiction to the conditions on (Q_{Λ}, R_{Λ}). 2\underline{nd} case: S ≅ S̸_{i} (i.e, i≠j). In this case, there is no oriented cycle in Q_{Λ} starting at i which is not in R_{Λ}. Then, from the condition (SP)4 and from dim_{k}P (j) = dim_{k}P_{i}(j)≥ 2, it follows that there is an oriented cycle μ∉R_{Λ}, starting at j, such that i is not a vertex of μ. So, relative to the paths with origin at i, we have one of the remaining cases of lemma 4.2, that is, (a.2.ii), (b.3.ii) or (b.4.ii). In case (a.2.ii), μ is a proper subpath of w_{α} and j is the end of two arrows γ and γ′, with γ′ in μ, and j is the origin of at least one arrow δ′, with δ′ in μ, such that μγ ∉R_{Λ} and δ′γ′ ∈ R_{Λ} (if there is another arrow δ in w_{α}, starting at j, then δμγ ∉ R_{Λ} and δγ ∈ R_{Λ}). From this, it follows that μ is the unique oriented cycle starting at j, and so μ = ˆu_{δ′} (since δ′γ′ ∈ R_{Λ}). In case (b.3.ii), μ is a proper subpath of exactly one of the parallel paths w_{α} and w_{β}, let us say w_{α}, and j is not the end of any of them. Therefore, j is the end of two distinct arrows γ and γ′ of w_{α} and the origin of two distinct arrows δ and δ′ as in (a.2.ii), and hence, μ = ˆu_{δ′}. If we have case (b.4.ii), since (w_{α})_{1} ∩(w_{β})_{1} =∅ and (w_{α})_{0} ∩(w_{β})_{0})={i}, we infer that μ is a proper subpath of only one of them, and relative to the arrows which have vertices at j it behaves as in case (a.2.i). Hence, in any of these possibilities, as in the first case, Q_{Λ} contains an oriented cycle, not in R_{Λ}, whose arrows are all different and such that no subpath of it is a summand of a generator of R_{Λ}, which gives a contradiction to condition (iii) of proposition 3.1. So, any projective indecomposable Λ-module is multiplicity-free.

Remark 4.2. In fact, the proof of theorem 4.3 follows from the fact that any oriented cycle of kQ_{Λ} is in R_{Λ}. Therefore if in this theorem we require that all oriented cycles are in the relation set instead of Λ-ind being determined by composition factors, its statement is still true.

To attain our aim, we need some preliminaries. Given a simple module S, we denote by i_{S} the vertex of Q_{Λ} which is associated to S. We start with another technical lemma.

Lemma 4.4. Let Λ be a representation-finite biserial algebra whose ordinary quiver Q_{Λ} has no loop and such that all oriented cycles are in R_{Λ}. Let N be a Λ-module, with l(N)≥2, and let v ∈N be such that the simple Λ-module S =kv ⊂ N is not a direct summand of N. Then

(i) There is a path

[figure: y_{r}; y_{r−1}; y_{2}; y_{1}; y_{0} = i_{S}; σ_{1}]^{σ_{r}}  ^{σ_{2}}  , r ≥ 1,

and a family (v_{i})_{i} ∈ ⨁_{i=0}^{r} N(y_{i}), with v_{0} = v, such that N(y_{i}) = kv_{i} ⊕ N′(y_{i}), N(σ_{i})(v_{i}) = v_{i−1} and N(σ_{i})(N′(y_{i})) ⊂ N′(y_{i−1}), for 1 ≤ i ≤ r, and Im N(δ) ⊂ N′(y_{r}), for any arrow δ ending at y_{r}.

(ii) If there are two arrows σ : y → i_{S} and \overline{σ} : \overline{y} → i_{S}, with σ ≠ \overline{σ}, such that v ∈Im N(σ) ∩ Im N(\overline{σ}), then the quiver Q_{N} of N contains the subquiver

[figure: y_{r}; y_{r−1}; y_{2}; y_{1}; y_{0} = i_{S}; σ_{1}] \overline{y}_{1}  \overline{y}_{2}  \overline{y}_{m−1}  \overline{y}_{m}^{\overline{σ}^{1}}^{σ_{2}}^{σ_{r}}  ^{\overline{σ}^{2}}  ^{\overline{σ}^{r}} ,

with r, m≥1, σ_{1} =σ, \overline{σ}_{1} =\overline{σ}. For this subquiver there are the families

(v_{i})_{i} ∈⨁_{i=0}^{r} N(y_{i}) and (\overline{v}_{j})_{j} ∈⨁_{j=0}^{m} N(\overline{y}_{j}),

where \overline{v}_{0} =v =v_{0}, such that N(y_{i})=kv_{i}⊕N′(y_{i}), N(σ_{i})(v_{i})=v_{i−1}, N(σ_{i})(N′(y_{i})) ⊂ N′(y_{i−1}), 1 ≤ i ≤ r, N(\overline{y}_{i}) = k\overline{v}_{i} ⊕ N′(\overline{y}_{i}), N(\overline{σ}_{i})(\overline{v}_{i}) = \overline{v}_{i−1}, N(\overline{σ}_{i})(N′(\overline{y}_{i})) ⊂ N′(\overline{y}_{i−1}), 1 ≤ i ≤ m. Furthermore, Im N(δ) ⊂ N′(y_{r}), for all arrows δ ending at y_{r}, and Im N(\overline{δ}) ⊂ N′(\overline{y}_{m}), for all arrows \overline{δ} ending at \overline{y}_{m}.

Proof. It follows, from remark 4.2, that each injective indecomposable Λ-module I is of type A_{n} or a projective-injective non-uniserial. Therefore, I(x) ≅ k for all x∈supp I, and I(α) ≅ 1 1 for every arrow α of the quiver Q_{I}.

We consider N(i_{S}) = kv ⊕ N′(i_{S}). Let ι: S ↪ N be the inclusion of S = kv in N and ȷ: S ↪ I_{S} be the injective envelope of S. Then, there exists a morphism of k-representations λ=(λ_{x})_{x}: N → I_{S} such that λ ∘ ι=ȷ. Since I_{S}(i_{S})=kv, we have that kerλ_{i_{S}} = N′(i_{S}). So, since S is not a direct summand of N, L = Imλ is a submodule of I_{S} containing S properly, and hence it is an indecomposable module with l(L) ≥ 2. Since I_{S} is of type A_{n} or a projective-injective non-uniserial, there is a subpath μ: y_{r} →^{σ_{r}} y_{r−1} … y_{2} →^{σ_{2}} y_{1} =y →^{σ_{1}} y_{0} =i_{S}, r ≥1, in the quiver of L, such that (Im λ_{y_{i}})(σ_{i}) ≅ 1 1 for 0 ≤ i ≤ r, and λ_{x} = 0 for each vertex x = o(δ), for every arrow δ, where e(δ) = y_{r}. It means that μ is the path of maximal length of Q_{L} ending at σ_{1} with the property L(μ) ≅ 1 1. Moreover, for each i = 1, 2, ..., r, since λ_{y_{i}} ≠0, the following diagram is commutative:

[figure: N(y_{i−1}); N(y_{i}); I_{S}(y_{i}) ≅ k; I_{S}(y_{i−1}) ≅ k; λ_{y_{i}}; λ_{i_{y_{i−1}}}; N(σ); ≅ 11]

From the commutativity of the above diagram and from L ≅ N/kerλ, we infer that we can choose a family (v_{i})_{i} ∈ ⨁_{i=0}^{r} N(y_{i}), where v_{0} = v, such that N(y_{i}) = kv_{i} ⊕ N′(y_{i}), where N′(y_{i}) = kerλ_{y_{i}}, and such that N(σ_{i})(v_{i}) = v_{i−1} and N(σ_{i})(N′(y_{i})) ⊂ N′(y_{i−1}), for 1 ≤ i ≤ r. Moreover, it follows easily that Im N(δ) ⊂ ker λ_{y_{r}} =N′(y_{r}), for each δ of Q_{N}, where e(δ)=y_{r} (since λ_{x} =0, where x=o(δ)). So, (i) is proved.

Let σ : y → i_{S} and \overline{σ} : \overline{y} → i_{S} be the two arrows of Q_{Λ}, with σ ≠ \overline{σ}, such that v ∈Im N(σ) ∩ Im N(\overline{σ}). Since Q_{Λ} does not have loops or double arrows, then \overline{y} ∉ \{y, i_{S}\}. In this case, I_{S} is a non-uniserial module, and it follows from this that l(L)≥3 and that λ_{y} and λ\overline{y} are not zero. So, L is a non-uniserial submodule of I_{S},

whose quiver is y_{r} →^{σ_{r}} y_{r−1} … y_{2} →^{σ_{2}} y_{1} =y →^{σ_{1}} y_{0} =i_{S} ←^{\overline{σ}_{1}} \overline{y}_{1} =\overline{y} ←^{\overline{σ}_{2}} \overline{y}_{2} … \overline{y}_{m−1} ←^{\overline{σ}_{m}} \overline{y}_{m},

where r, m≥1, σ_{1} =σ, \overline{σ}_{1} =\overline{σ}. (We observe that, if \overline{y}_{m} =y_{r}, then I_{S} is an injectiveprojective non-uniserial and λ is an epimorphism). The rest of the proof follows using arguments as in part (i) on each of the two paths ending at i_{S}, which make Q_{L}.

Remark 4.3. Under the hypothesis of lemma 4.4, if there are two different arrows σ: y → i_{S} and \overline{σ}: \overline{y} → i_{S} of Q_{Λ} such that v ∈Im N(σ) but v ∉Im N(\overline{σ}), then, by a conveniently chosen basis for N(\overline{y}), we infer that Im N(\overline{σ}) ⊂ N′(i_{S}). Furthermore, using (SP)2, we can describe the morphism N(α), for each arrow α different from σ_{i}, for 1 ≤ i ≤ r, having one of its vertices at y_{i}, 0 ≤ i ≤ r. If α ≠ σ_{i}, 1 ≤ i ≤ r − 1, with o(α) ∈ {y_{i}, i_{S}}, then N(α)(v_{i}) = 0 for 1 ≤ i ≤ r; if α ≠ σ_{i+1}, 1 ≤ i ≤ r − 1, with e(α) = y_{i}, then Im N(α) ⊂ N′(y_{i}). Even in the case that v ∈ Im N(σ) ∩ Im N(\overline{σ}), we can also describe similarly the morphisms N(γ), for each γ ≠\overline{σ}_{j}, 1≤j ≤m, and having one of its vertices at \overline{y}_{j}, for 1≤j ≤m − 1.

The last lemma and remark 4.3 allow us to determine if a given module M is indecomposable or not through of the number of arrows ending at the vertices of the quiver of M, which are associated to their simple submodules.

Theorem 4.5. Let Λ=k(Q_{Λ}, R_{Λ}) be a representation-finite biserial algebra whose ordinary quiver Q_{Λ} has no loop and such that all oriented cycles of Q_{Λ} are in R_{Λ}. Let M be a Λ-module such that M = N + X, with l(N) , l(X) ≥ 2, and suppose that there exists 0 ≠ v ∈ M such that N ∩ X = kv = S. If there is an arrow σ ending at i_{S} such that v ∈Im N(σ) ∩ Im X(σ), then M is decomposable.

Proof. Since S = kv ⊂ M, it follows that M(α)(v) = 0 for each arrow α where o(α)=i_{S}. On the other hand, setting N(i_{S})=kv ⊕ N′(i_{S}) and X(i_{S})=kv ⊕ X′(i_{S}), we can rewrite M =(M(x)_{x}, M(α)_{α}) in the following way:

M(x) =\{ \begin{matrix}N(x) ⊕ X(x) & if x≠i_{S} , \\ kv ⊕ N′(i_{S}) ⊕ X′(i_{S}) & if x = i_{S}, \end{matrix}

M(α) =\{ \begin{matrix}N(α) ⊕ X(α) & if i_{S} ∉\{o(α), e(α)\}, \\ N(α) + X(α) & if e(α)=i_{S}, \\ N′(α) ⊕ X′(α) & if o(α)=i_{S}, \end{matrix}

where X′(α) and N′(α) denote, respectively, the restriction of X(α) to the subspace X(o(α)) and of N(α) to the subspace N(o(α)). It is clear that S is neither a direct summand of N nor of X. We consider several cases.

Let us assume first that, besides the arrow σ: y → i_{S}, there is in Q_{M} another arrow \overline{σ}: \overline{y} → i_{S}, where \overline{σ} ≠ σ (hence y ≠ \overline{y}) and that v ∈ Im N(\overline{σ}) ∩ Im X(\overline{σ}). Applying lemma 4.4(ii) to N and X, we see that there is a subquiver μ_{N} of Q_{N},

where μ_{N}: y_{r} →^{σ_{r}} … → y_{s} →^{σ_{s}} … → y_{2} →^{σ_{2}} y_{1} = y →^{σ_{1}} y_{0} = i_{S} ←^{\overline{σ}_{1}} \overline{y}_{1} = \overline{y} ←^{\overline{σ}_{2}} \overline{y}_{2} ←

… ←^{\overline{σ}_{m}} \overline{y}_{m}, and there is a subquiver μ_{X} of Q_{X}, where μ_{X}: y_{s} →^{σ_{s}} … → y_{2} →^{σ_{2}}

y_{1} = y →^{σ_{1}} y_{0} = i_{S} ←^{\overline{σ}_{1}} \overline{y}_{1} = \overline{y} ←^{\overline{σ}_{2}} \overline{y}_{2} ← … ← \overline{y}_{q−1} ←^{\overline{σ}_{q}} \overline{y}_{q}, where 1 ≤ s ≤ r, and

q, m ≥ 1, σ_{1} = σ, \overline{σ}_{1} = \overline{σ}. For these subquivers, also by lemma 4.4(ii), there are families (v_{i})_{i} ∈ ⨁_{i=0}^{r} N(y_{i}), (\overline{v}_{i})_{i} ∈ ⨁_{i=0}^{m} N(\overline{y}_{i}), and (w_{j})_{j} ∈ ⨁_{j=0}^{s} X(y_{j}) and (\overline{w}_{j})_{j} ∈ ⨁_{j=0}^{q} , X(\overline{y}_{j}), where v_{0} = w_{0} = \overline{v}_{0} = \overline{w}_{0} = v, such that N(y_{i}) = kv_{i} ⊕ N′(y_{i}), for 0 ≤ i ≤ r, N(\overline{y}_{i}) = k\overline{v}_{i} ⊕ N′(\overline{y}_{i}), for 0 ≤ i ≤ m, and X(y_{j}) = kw_{j} ⊕ X′(y_{j}), for 0 ≤ j ≤ s and X(\overline{y}_{j}) = k\overline{w}_{j} ⊕ X′(\overline{y}_{j}), for 0 ≤ j ≤ q. Moreover, the morphisms X(α) and N(α) are as in lemma 4.4 and in remark 4.3, for the arrows α starting (ending) at y_{i} and \overline{y}_{j}.

Since all the paths of kQ_{Λ} − R_{Λ} ending in \overline{σ} are ordered by their lengths, it follows that q ≤ m or m ≤ q. Suppose now that 1 ≤ m ≤ q. Using the previous notation, let L=(L(x)_{x}, L(α)_{α}) be defined by

L(x)=\begin{cases}kv_{i} ⊕ N′(y_{i}) ⊕ X′(y_{i}) & if x=y_{i}, for 0≤i≤s, \\ kv_{i} ⊕ N′(y_{i}) ⊕ X(y_{i}) & if x=y_{i}, for s + 1≤i≤r, \\ N′(\overline{y}_{j}) ⊕ k\overline{w}_{j} ⊕ X′(\overline{y}_{j}) & if x=\overline{y}_{j}, for 1≤j ≤m, \\ N(\overline{y}_{j}) ⊕ k\overline{w}_{j} ⊕ X′(\overline{y}_{j}) & if x=\overline{y}_{j}, for m + 1≤j ≤q, \\ N(x) ⊕ X(x) & for the other vertices, \end{cases}

L(α)=\begin{cases}N(σ_{i}) ⊕ X′(σ_{i}) & if α=σ_{i}, for 1≤i≤s, \\ N′(\overline{σ}_{i}) ⊕ X(\overline{σ}_{i}) & if α=\overline{σ}_{i}, for 1≤i≤m, \\ N(α) ⊕ X(α) & for the other arrows. \end{cases}

It is easy to verify that L is a representation of (Q_{Λ}, R_{Λ}). Let ı=(ı_{x})_{x}: L → M be a family where, for each x, ı_{x}: L(x) → M(x) is the natural inclusion from the space L(x) to the space M(x). It is also easy to prove that it admits a left inverse morphism p = (p_{x})_{x}: M → L. The components p_{x} of p are the following. For 1 ≤ i ≤ s, p_{y_{i}}: kv_{i} ⊕ N′(y_{i}) ⊕ kw_{i} ⊕ X′(y_{i}) → kv_{i} ⊕ N′(y_{i}) ⊕ X′(y_{i}) is defined by p_{y_{i}}(av_{i} +n′ +bw_{i} +x′)=(a+b)v_{i} +n′ +x′, with a, b∈ k, x′ ∈X′(y_{i}) and n′ ∈N′(y_{i}); for 1≤ i≤ m, p_{\overline{y}_{i}}: k\overline{v}_{i} ⊕ N′(\overline{y}_{i}) ⊕ k\overline{w}_{i} ⊕ X′(\overline{y}_{i}) → N′(\overline{y}_{i}) ⊕ k\overline{w}_{i} ⊕ X′(\overline{y}_{i}) is defined by p_{\overline{y}_{i}}(a\overline{v}_{i} + n′ + b\overline{w}_{i} + x′) = (a + b)\overline{w}_{i} + n′ + x′, with a, b ∈ k, n′ ∈ N′(\overline{y}_{i}) and x′ ∈ X′(\overline{y}_{i}) and, for the remaining vertices x, p_{x} : N(x) ⊕ X(x) → N(x) ⊕ X(x) is the identity. We leave the verification to the reader.

For the remaining cases (that is, either v ∈ Im N(\overline{σ}) ∩ Im X(\overline{σ}) and q ≤ m, or v ∉ Im N(\overline{σ}) ∩ Im X(\overline{σ}), or σ is the unique arrow of Q_{M} ending at i_{S} such that v ∈ Im N(σ) ∩ Im X(σ)), by the application of lemma 4.4 and remark 4.3 and a slight modification of the above arguments, it is not difficult to verify that M =N ⊕ X/kv. So the theorem is proved.

Corollary 4.6. Let Λ = k(Q_{Λ}, R_{Λ}) be a representation-finite biserial k-algebra whose ordinary quiver Q_{Λ} does not have loops and such that all oriented cycles of Q_{Λ} are in R_{Λ}. Let M be a Λ-module such that M=N+X, where l(N), l(X) ≥ 2 and N ∩ X = S = kv, for some 0 ≠ v ∈ M. If M is indecomposable, then there is a unique arrow σ : y → i_{S} such that v ∈ Im N(σ), and a unique arrow \overline{σ} : \overline{y} → i_{S}, where σ ≠\overline{σ}, such that v ∈Im X(\overline{σ}).

Proof. The proof follows immediately from the fact that M is indecomposable and from lemma 4.4 and theorem 4.5.

Theorem 4.7. Let Λ= k(Q_{Λ}, R_{Λ}) be a representation-finite biserial algebra whose ordinary quiver Q_{Λ} has no loop and such that all oriented cycles of kQ_{Λ} are in R_{Λ}. Let M be a Λ-module and S be a simple Λ-module such that S ⊂ M and M/S = M_{1} ⊕ M_{2} ⊕ … ⊕ M_{t}, where t ≥ 1 and each M_{j} is indecomposable, for j =1, 2, …, t. If M is indecomposable, then t≤2.

Proof. Let 0 ≠ v ∈ M be such that S = kv. We consider now the canonical epimorphism π: M → M/S. For each j = 1, 2, ..., t, we put N_{j} = π^{−1}M_{j}, the reciprocal image of M_{j} under π. It is clear that M =∑_{j=1}^{t} N_{j} and N_{i} ∩ N_{j} =S, if i≠j. Moreover, N_{j}/S = M_{j}, for j = 1, 2, ..., t. It is clear that each N_{j} is an indecomposable module.

Suppose now that t≥3. If we write M =N_{1} + X, where X =∑_{j=2}^{t} N_{j}, it follows that N_{1}∩X =S =kv and, by the corollary above, there is a unique arrow σ, ending at i_{S}, such that v ∈Im N_{1}(σ), and there is a unique arrow \overline{σ}, ending at i_{S}, with \overline{σ} ≠σ, such that v ∈ Im X(\overline{σ}). So, applying lemma 4.4(i) to each N_{j}, for j = 2, 3, ..., t, and recalling that Im X(\overline{σ}) = ∑_{j=2}^{t} Im N_{j}(\overline{σ}), we see that v ∈ Im N_{j}(\overline{σ}), for all j = 2, 3, ..., t. On the other hand, if we write X = N_{2} + (N_{3} + ... + N_{t}), then from theorem 4.5 it follows that X = X_{1} ⊕ X_{2}, where X_{i} ≠ (0) for i = 1, 2, and S ⊂ X_{1}. Therefore, M = N_{1} + (X_{1} ⊕ X_{2}) = X_{2} + (N_{1} + X_{1}); since M/S = M_{1} ⊕ M_{2} ⊕ ... ⊕ M_{t}, and N_{j} =π^{−1}M_{j}, for j =1, 2, ..., t, then X_{2} ∩ (N_{1} + X_{1})=(0). This means that M =X_{2} ⊕ (N_{1} + X_{1}), which is a contradiction to the assumption that M is indecomposable.

We observe that, given a module M and a simple submodule S of M, the results which we have gotten until now rely on the existence of paths in the quiver of M ending at the vertex corresponding to S. By duality, we can get the dual results about the simple modules T which are direct summands of top M.

5. The multiplicity of the composition factors of some modules

We are now going to measure the multiplicity of a simple submodule S of a given module M, when the quotient M/S is of type A_{n} for some n ≥ 2, and also when M/S is a direct sum of two submodules of type A_{n}, in case Λ is a biserial algebra whose ordinary quiver has no loop and Λ-ind is determined by composition factors. Dually, we can measure the multiplicity of a simple module T as a composition factor of M, where there is an epimorphism from M to T whose kernel is of type A_{n} or is a direct sum of two submodules of type A_{n}.

We first need to introduce some notations and conventions which make reading easier. Let Q′ be a quiver whose underlying graph is a diagram A_{n}, for n ≥ 3. A vertex of Q′ which belongs to exactly two arrows will be called an interior vertex of Q′; and a vertex of Q′ which belongs to exactly one arrow will be called an extreme vertex of Q′. So, all vertices of Q′, except two of them, are interior vertices. The extreme vertices can be distinguished in the following way: if x is the origin of the unique arrow of Q′ to which it belongs, we call x the initial extreme; otherwise, we call it the terminal extreme. On the other hand, the interior vertices can be divided into three classes. If x is an interior vertex such that it is the origin of the two arrows to which it belongs, we call x a 2-source. If x is the end of the two arrows to which it belongs, we call x a 2-sink. In the remaining case, x is called a transit point.

We now fix the following convention: Let M′ be a module of type A_{n}, with n≥2. If the quiver of M′ contains a proper subquiver whose orientation is not important for the context, we shall represent this subquiver by points (vertices) and by edges.

Using the notations and conventions above, we have:

Lemma 5.1. Let Λ = k(Q_{Λ}, R_{Λ}) be a representation-finite biserial algebra whose ordinary quiver Q_{Λ} has no loop. Let M be a Λ-module whose quiver is connected, and S = kv ⊂ M be a simple submodule, where 0 ≠ v ∈ M. Suppose that \overline{M} = M/S is of type A_{n}, n≥2, and that there is \overline{v} ∈\overline{M}(i_{S}) such that \overline{S} =k\overline{v} ⊂ \overline{M}. Then M is one of the following k-representation of (Q_{Λ}, R_{Λ}), relative to its support:

[figure: kv ⊕ k\overline{v}; ^{( \begin{matrix}λ \\ 1 \end{matrix} )}; k; …; A.1; (λ=0 or λ ≅ 1 1); or; kv ⊕ k\overline{v}; k; ^{( \begin{matrix}λ_{1} \\ 1 \end{matrix} )}; k; ^{( \begin{matrix}λ_{2} \\ 1 \end{matrix} )}; …; A.2; (λ_{i} =0, i=1, 2, or λ_{i} ≅ 1 1, i=1, 2)]

Moreover, S is a direct summand of M.

Proof. From the assumptions on \overline{M}, we write M(i_{S})=kv ⊕ k\overline{v} and M(x)=\overline{M}(x) ≅ k, for all x ∈ supp M, x ≠ i_{S}. Since S = kv ⊂ soc M and k\overline{v} ⊂ \overline{M}, it follows that M(δ)(v) = \overline{M}(δ)(v) = 0 for each arrow δ, where o(δ) = i_{S}, and M(α) = \overline{M}(α), for each arrow α of Q_{Λ}, with i_{S} ∉\{o(α), e(α)\}. So, the quiver of \overline{M} and, consequently, the quiver of M does not contain arrows starting at i_{S}. Therefore, we conclude that i_{S} is a sink in Q_{M}. Since the quiver Q_{\overline{M}} is obtained from the quiver Q_{M} by eventual exclusion of the arrows to which i_{S} belongs, we need to analyse the neighborhood of i_{S} in Q_{\overline{M}}. 1\underline{st} case: i_{S} is a 2-sink in Q_{\overline{M}}. Under this assumption, we infer that the quivers Q_{M} and Q_{\overline{M}} are equal and i_{S} is a 2-sink of Q_{M}. Denote by σ_{1} and σ_{2}, where σ_{1} ≠σ_{2}, arrows of Q_{M} such that e(σ) = i_{S} for i = 1, 2. Then, for i = 1, 2, M(σ_{i}) ≅ ( \begin{matrix}λ_{i} \\ 1 \end{matrix} ), where λ_{i} = 0 or λ_{i} ≅ 1 1. Suppose that λ_{1} =0 and λ_{2} ≅ 1 1, and let M_{1} and M_{2} be the k-subrepresentations of M, which are represented, relative to its support, by

M_{1}  M_{2} [figure: k\overline{v}; 1; k; k; kv; 1; …]

These k-representations are such that k\overline{v} ⊂ soc M_{1} and kv ⊂ soc M_{2}. Consider M_{1} ⊕ M_{2}, which is represented by

[figure: kv ⊕ k\overline{v}; k; ^{( \begin{matrix}0 \\ 1 \end{matrix} )}; k; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; …]

The family f = (f_{x})_{x}: M ⟶ M_{1} ⊕ M_{2}, given by f_{x} = 1 1_{M(x)} if x ≠ i_{S} and fi_{S} (av +b\overline{v})=av +(b−a)\overline{v} for a, b∈k, defines an isomorphism of k-representations. Hence, M/S ≅ M_{1} ⊕ M_{2}/kv, and it is not of type A_{n}. Therefore, λ_{i} = 0 for each i=1, 2, or λ_{i} ≅ 1 1 for each i=1, 2, and M is one of the representation in A.2. In the first case, it is very simple to verify that S is a direct summand of M. In the second case, we have that the natural inclusion from kv in M is a monomorphism which admits as a left inverse morphism of k-representations the family p=(p_{x})_{x}: M ⟶ S =kv, defined by by p_{x} =0 if x≠i_{S} and pi_{S} (av + b\overline{v})=(a − b)v for a, b∈k. 2\underline{nd} case: i_{S} is a terminal extreme in Q_{\overline{M}}. Let σ: y → i_{S} be the unique arrow of Q_{\overline{M}} to which i_{S} belongs. Then M(σ) = ( \begin{matrix}λ \\ 1 \end{matrix} ), where λ: k → kv is zero or is a

scalar multiple of the identity. Since Q_{Λ} does not contain subquivers of type Ã_{m} without relations and verifies (SP)2, and since supp M =supp \overline{M}, it is easy to verify that σ is the unique arrow of Q_{Λ}, ending at i_{S}, which is an arrow of Q_{M}. This implies that the quivers Q_{M} and Q_{\overline{M}} coincide, and we have case A.1. By the same arguments used in the first case, we obtain that the natural inclusion from S = kv to M splits, that is, S is a direct summand of M.

In the following lemma, we consider a k-representation M whose top admits a composition factor isomorphic to a direct summand of soc M.

Lemma 5.2. Let Λ = k(Q_{Λ}, R_{Λ}) be a representation-finite biserial algebra whose ordinary quiver has no loop. Let M be a Λ-module whose quiver Q_{M} is connected. Consider v ∈M, v ≠0, and S =kv ⊂ M, a simple module. Suppose that \overline{M} =M/S is of type A_{n}, n≥2. If there is 0≠\overline{v} ∈\overline{M}(i_{S}) such that there exists an epimorphism from \overline{M} to \overline{S} =k\overline{v}, then M is one of the following k-representations, relative to its support:

[figure: …; k; kv ⊕ k\overline{v}; ^{(^{0 1} )}; B.1; …; k; kv ⊕ k\overline{v}; ^{(^{0 1} )}; ^{(^{0 1} )}; k; B.3]

[figure: …; k; kv ⊕ k\overline{v}; ^{(^{0 1} )}; k; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; 1; k; B.2; …; k; kv ⊕ k\overline{v}; ≅^{(^{0 1})}≅; k; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; 1; k; k; B.4]

[figure: …; k; kv ⊕ k\overline{v}; ≅^{(^{0 1})}≅; k; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; 1; k; k; …; k; k; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; 1] B.5

Furthermore, in cases B.1 and B.3, S =kv is a direct summand of M.

Proof. As before, we can write M(i_{S}) = kv ⊕ k\overline{v} and M(x) = \overline{M}(x) ≅ k, for all vertex x≠i_{S} of Q_{M}, and M(α)=\overline{M}(α) for all arrows α with i_{S} ∉\{o(α), e(α)\}, and M(δ)(v) = 0 for each arrow δ with o(δ) = i_{S}. From the hypothesis on \overline{M} and \overline{S}, as in the previous lemma, it is easy to see that i_{S} is a source of Q_{\overline{M}}. Moreover, from the epimorphism M → \overline{M} → k\overline{v}, it follows that k\overline{v} ∩ ∑_{γ} Im M(γ) = (0), for each arrow γ, where e(γ) = i_{S}. We have now some cases, according to which kind of source i_{S} is of Q_{\overline{M}} and if there is an arrow ending at i_{S} in Q_{M}. 1\underline{st} case: i_{S} is a 2-source of Q_{\overline{M}}. Consider δ_{1} : i_{S} → t_{1} and δ_{2} : i_{S} → t_{2}, where δ_{1} ≠ δ_{2} (hence, t_{1} ∉ \{i_{S}, t_{2}\}), such that \overline{M}(δ_{1}) ≅ 1 1 ≅ \overline{M}(δ_{2}). So, M(δ_{1}) ≅ ( ^{0} ^{1} ) ≅ M(δ_{2}). According to whether i_{S} is an end point of arrows in Q_{M}, we have the following subcases.

1.1: There is no arrow in Q_{M} ending at i_{S} (i.e, i_{S} is a 2-source of Q_{M}). This condition implies that the quivers Q_{M} and Q_{\overline{M}} coincide, and so M is the k-representation in B.3.

1.2: There is in Q_{M} exactly one arrow γ with e(γ)=i_{S}. Since supp M =supp \overline{M}, \overline{M} is of type of A_{n} and the condition (SP)2 is satisfied, it follows that o(γ)= x≠i_{S} is an extreme vertex of Q_{\overline{M}}. Hence, since Q_{Λ} does not contain loops, double arrows or subquivers of type Ã_{r} without relations (since Λ is representation-finite), it results that there is, in Q_{\overline{M}}, a path from i_{S} to x (this path is unique and x is its terminal extreme). Then there exists an oriented cycle in Q_{M} containing this path and γ. On the other hand, since γ: x → i_{S} is exactly the one arrow of Q_{Λ} such that M(γ)≠0, it follows that k\overline{v} ∩ Im M(γ) = k\overline{v} ∩ ∑_{γ} Im M(γ) = (0), and so, by a convenient choice, we have that M(γ) ≅ ( \begin{matrix}1 \\ 0 \end{matrix} ), that is, M is the k-representation in B.4.

1.3: There are, in Q_{M}, two different arrows γ_{1} and γ_{2} with e(γ_{i})=i_{S}, for i=1, 2. Let x_{i} = o(γ_{i}), for i = 1, 2. Then, x_{1} ≠ x_{2} and i_{S} ∉ {x_{1}, x_{2}}. As in 1.2, for each i = 1, 2, x_{i} is an extreme vertex of Q_{\overline{M}} and, hence, each x_{i} is the terminal vertex of the unique path in Q_{\overline{M}} joining i_{S} to x_{i}. So, this means that Q_{\overline{M}} is the union of these paths and, consequently, Q_{M} is the union of the two oriented cycles which are determined by the mentioned paths above and the arrows γ_{i}, for i=1, 2, having a unique common vertex, the vertex i_{S}. As in case 1.2, by a convenient choice, we have that M(γ_{i})= ( \begin{matrix}1 \\ 0 \end{matrix} ) for i=1, 2, and, hence, M is the k-representation in B.5.

2\underline{nd} case: i_{S} is an initial extreme of Q_{\overline{M}}. Let δ: i_{S} → t be the unique arrow such that \overline{M}(δ) ≅ 1 1. It is clear that δ is the unique arrow of Q_{M} starting at i_{S}, and so, M(δ) ≅ ( ^{0} ^{1} ). By the analysis of whether the Q_{M} contains arrows ending at i_{S}, it is easy to see, using analogous arguments, that either Q_{M} and Q_{\overline{M}} coincide (and, in this case, M is the k-representation in B.1), or Q_{\overline{M}} is a proper subpath of Q_{M}, obtained by elimination of one arrow ending at i_{S}. In this last case the quiver of M is an oriented cycle and hence M is the representation in B.2.

In both cases B.1 and B.3, it is very simple to verify that the natural inclusion of S =kv in M is a splittable monomorphism.

Lemma 5.3. Let Λ = k(Q_{Λ}, R_{Λ}) be a biserial representation-finite type algebra whose ordinary quiver Q_{Λ} has no loop. Let M be a k-representation whose quiver is connected. Consider v ∈ M and the simple submodule of M, S = kv. Suppose that \overline{M} = M/S is of type A_{n}, n ≥ 2. If there is 0 ≠ \overline{v} ∈ \overline{M}(i_{S}) such that the simple module \overline{S} = k\overline{v} is not a direct summand of the module soc \overline{M} ⊕ top \overline{M}, then M is one of the following k-representations:

[figure: k; ^{(^{0 1} )}; kv ⊕ k\overline{v}; ^{( \begin{matrix}λ \\ 1 \end{matrix} )}; …; k; or; C.1; (λ=0 or λ ≅ 1 1); k; ^{(^{0 1} )}; kv ⊕ k\overline{v}; ^{( \begin{matrix}0 \\ 1 \end{matrix} )}; …; k; …; k; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; C.2]

In case C.1, S =kv is a direct summand of M.

Proof. The proof of this lemma is very similar to the proof of the previous lemmas, and we leave it for the reader.

Proposition 5.4. Let Λ = k(Q_{Λ}, R_{Λ}) be a representation-finite biserial k-algebra such that the ordinary quiver Q_{Λ} does not have any loop. Let M be an indecomposable Λ-module and S be a simple submodule of M such that \overline{M} =M/S is of type A_{n}, with n ≥ 3. If \overline{M}(i_{S}) is non-zero, then M is one of the k-representations in B.2, B.4, B.5 in lemma 5.2 or C.2 in lemma 5.3.

Proof. Let 0 ≠ v ∈ M be such that S = kv and 0 ≠ \overline{v} ∈ \overline{M}(i_{S}). Consider the simple module k\overline{v}. Since k\overline{v} is a composition factor of \overline{M} and Q_{M} is connected (since M is indecomposable), by lemma 5.1, k\overline{v} is a composition factor of top M or is not a composition factor of soc \overline{M} ⊕ top M. If k\overline{v} is a composition factor of top M, from lemma 5.2 it follows that M is one of the representations in B.2, B.4 or B.5. If, instead, k\overline{v} is not a composition factor of the module soc \overline{M} ⊕ top M, then it is not a composition factor of the module soc \overline{M} ⊕ top \overline{M}, and from lemma 5.3, it follows that M is the representation in C.2.

Proposition 5.5. Let Λ be a biserial algebra whose ordinary quiver has no loop. Suppose that the category Λ-ind is determined by composition factors. Let M be an indecomposable Λ-module, where l(M) ≥ 3, and S be a simple Λ-module such that S ⊂ M and \overline{M} =M/S is of type A_{n}. Then M is multiplicity-free.

Proof. Since Λ-ind is determined by composition factors, then by remark 1.1 Λ is a representation-finite biserial algebra and, therefore, Λ ≅ k(Q_{Λ}, R_{Λ}), under the conditions of the proposition and the conditions of remarks 4.1 and 4.2. Under the assumption on M and \overline{M}, we have that M(x) ≅ \overline{M}(x) ≅ k, for all x ∈ supp M, x≠i_{S}. To get the result we need to prove that \overline{M}(i_{S})=0. Suppose that \overline{M}(i_{S}) ≅ k. By proposition 5.4, M is one of the k-representations in B.2, B.4, B.5 or C.2. In any of these cases, it is immediate to see that there is a subrepresentation N_{1} of \overline{M} ≅ M/kv and a subrepresentation N_{2} of M, where l(N_{i}) ≥ 2, for each i = 1, 2, which are represented, relative to its support, by

[figure: k; 1; k\overline{v}; …; k; N_{1}; k; 1; kv; …; k; N_{2}]

where the dotted arrows indicate that such arrows do not belong to the quiver of N_{i}, for each i = 1, 2 (that is, the k-morphisms to which they correspond are zero). The k-representations N_{1} and N_{2} are of type A_{r}, where r = l(N_{1}) = l(N_{2}), and so, they are indecomposable modules having the same composition factors, but they are not isomorphic (since soc N_{2} ≅ kv ≅ s̸oc N_{1}). We obtain a contradiction to the assumption that Λ-ind is determined by composition factors. Therefore, M(x) ≅ k, for all vertices of the quiver of M, that is, M is multiplicity free.

Proposition 5.6. Let Λ be a biserial k-algebra whose ordinary quiver does not have any loop and such that Λ-ind is determined by composition factors. Let M be an indecomposable Λ-module, l(M) ≥ 3, and S be a simple submodule of M such that \overline{M} = M/S = N_{1} ⊕ N_{2}, where the N_{i} are of type A_{n_{i}}, n_{i} ≥ 1, for i = 1, 2. Then S is a composition factor of M with multiplicity one.

Proof. As in the proof of proposition 5.5, Λ ≅ k(Q_{Λ}, R_{Λ}) satisfies the same conditions mentioned there. Let 0≠v ∈M be such that S =kv. We consider the canonical epimorphism π: M → \overline{M} and we denote by M′_{j}, for j = 1, 2, the reciprocal image of N_{j} by π. Then, M = M′_{1} + M′_{2} and M′_{1} ∩ M′_{2} = kv. Moreover, for each j = 1, 2, M′_{j}(x)=N_{j}(x) ≅ k, where x∈supp N_{j}, x≠i_{S}, and M(i_{S})=kv ⊕ N_{1}(i_{S}) ⊕ N_{2}(i_{S}).

Suppose, for j = 1, that N_{1}(i_{S}) ≅ k. Then, M′_{1}(i_{S}) = kv ⊕ N_{1}(i_{S}) ≅ kv ⊕ k. On the other hand, since M is an indecomposable module, by corollary 4.6, there is a unique arrow σ: y → i_{S} such that v ∈ Im M′_{1}(σ); then l(M_{1}) ≥ 3 and the quiver of M′_{1} is connected. Since M′_{1}/S = N_{1} is of type A_{n_{1}}, it follows that M′_{1} is one of the representations in lemma 5.1, 5.2 or 5.3. But, since Λ-ind is determined by composition factors, from the proof of proposition 5.5, we infer that M′_{1} is one of the representations in A.1, A.2, B.1 or B.3. In any of these cases, S is a direct summand of M′_{1} and, consequently, a direct summand of M, which contradicts the indecomposability of M. So, M(i_{S}) = kv, that is, S has multiplicity one as a composition factor of M.

6. The main theorems

The methods developed in the previous sections allow us now to prove the following results.

Theorem 6.1. Let Λ=k(Q_{Λ}, R_{Λ}) be a representation-finite biserial algebra. Suppose that the ordinary quiver of Λ does not have any loop and that all oriented cycles of Q_{Λ} are in R_{Λ}. If M is a multiplicity-free indecomposable Λ-module, then M either is a projective-injective non-uniserial module or is of type A_{n}, where n=l(M).

Proof. We first observe that (Q_{Λ}, R_{Λ}) was chosen satisfying the conditions (SP) of theorem 4.1, and also the conditions of remark 1.1. Since M is a multiplicity-free indecomposable module, then M(x) ≅ k, for each x∈supp M. We shall proceed by induction on the length of the multiplicity-free indecomposable modules. It is clear that each indecomposable module M, where l(M)=n≤3, is of type A_{n}, since it is a local or colocal module.

Suppose now that M is an indecomposable multiplicity-free module, with l(M)= n≥4, and that the result is true for all indecomposable multiplicity-free module N such that 1 ≤ l(N) < l(M). Let S be a simple submodule of M and consider the quotient M/S. If M/S is indecomposable then, by the inductive assumption, M/S is of type A_{n−1}. Let {i_{1}, i_{2}, ..., i_{n−1}}=supp M/S be ordered in such a way that i_{1} and i_{n−1} are the extreme vertices of the quiver of M/S. Since S ⊂ M and M is an indecomposable multipliciy-free module (therefore, Q_{M} is connected), then there is at least one arrow σ: i_{j} → i_{S}, for some j, 1≤j ≤n − 1 (and at most two), such that σ is in Q_{M}. From the conditions on (Q_{Λ}, R_{Λ}), it follows that o(σ)∈{i_{1}, i_{n−1}}.

Suppose that the quiver of M/S is a path, for example a path from i_{n−1} to i_{1}. Since Q_{Λ} contains no subquiver of type Ã_{r} without relations or loops, then Q_{M} does not contain another arrow \overline{σ} ≠σ of Q_{Λ} ending at i_{S}. So, the underlying diagram of Q_{M} is of type A_{n}, and M is of type A_{n}, where n = l(M). Otherwise, if the quiver of M/S is of type A_{n−1}, but is not a path, then there is at least one vertex i_{m}, 2≤m≤n − 2, such that i_{m} is either a 2-source or a 2-sink of Q_{M/S}.

Let us assume first that i_{m} is the unique 2-source of Q_{M/S}. Then Q_{M/S} is the union of a path μ from i_{m} to i_{1} and a path ν from i_{m} to i_{n−1} in Q_{M/S}. So, we infer that either σ is the unique arrow of Q_{Λ} ending at i_{S} which is an arrow of Q_{M} or, besides σ, Q_{M} contains another arrow \overline{σ}, where e(\overline{σ})=i_{S}, o(σ)=i_{1} and o(\overline{σ})=i_{n−1} such that σμ − \overline{σ}ν ∈ R_{Λ} (by (SP)4). In the first alternative, it follows imediately that the underlying graph of Q_{M} is a diagram A_{n}, and so, M is of type A_{n}. The second alternative implies that Q_{M} is

[figure: i_{m}; i_{1}; i_{n−1}; i_{S}; σ; \overline{σ}]

and, consequently, M is a projective-injective non-uniserial module.

Otherwise, if i_{m} is not the unique 2-source or i_{m} is a 2-sink of Q_{M/S}, then σ is the unique arrow of Q_{Λ} ending at i_{S} which is an arrow of Q_{M}, since Q_{Λ} contains no subquiver of type Ã_{r} without relations. This implies that M is of type A_{n}.

Finally, we need to consider the case in which M/S is a decomposable module. By theorem 4.7, M/S =M_{1} ⊕ M_{2}, where, for each i=1, 2, M_{i} is an indecomposable non-projective module. Moreover, each M_{i} is multiplicity-free and they have disjoint supports. Denoting by n_{i} the length of M_{i}, i=1, 2, we have that M_{i} is of type A_{n_{i}} (by the induction assumption, because l(M_{i}) < l(M), i = 1, 2). If we now consider the canonical epimorphism π: M → M/S, we can write M =N_{1} + N_{2}, where N_{1} ∩ N_{2} =S and N_{i} is the reciprocal image of M_{i} under π. Then, since M(i_{S}) ≅ k and supp N_{1} ∩ supp N_{2} ={i_{S}}, by corollary 4.6, there is a unique arrow σ_{1} : y_{1} → i_{S} such that N_{1}(σ_{1}) ≅ 1 1 and there is a unique arrow σ_{2} : y_{2} → i_{S}, where σ_{2} ≠ σ_{1} and y_{1} ∉ {y_{2}, i_{S}}, such that N_{2}(σ_{2}) ≅ 1 1. Moreover, for each i = 1, 2, y_{i} ∈ supp M_{i} is a extreme vertex of the quiver Q_{M_{i}}. So, we conclude that Q_{M} has for its underlying graph a diagram A_{n}, and the theorem is proved.

Theorem 6.2. Let Λ be a connected, basic biserial k-algebra whose ordinary quiver does not have a loop. Suppose that Λ-ind is determined by composition factors. Then each indecomposable Λ-module is multiplicity-free.

Proof. Since Λ-ind is determined by composition factors, by remark 1.1, Λ is a representation-finite biserial algebra, and so, we choose Λ ≅ k(Q_{Λ}, R_{Λ}), where (Q_{Λ}, R_{Λ}) satisfies the conditions (SP) of theorem 4.1 and those of proposition 3.1. Moreover, according to remark 4.2, all oriented cycles of kQ_{Λ} are in R_{Λ}. We shall proceed by induction on the length of the indecomposable modules. Let M be in Λ-ind and such that l(M) ≤ 3. Is easy to verify that each indecomposable of length less than or equal to 3 is a local or colocal module, and, by theorem 4.3, it follows that M is multiplicity-free. Suppose that M is indecomposable such that l(M) ≥ 4 and that the result is true for each indecomposable module N such that 1≤l(N)<l(M). By duality it is enough to prove that if S is a composition factor of M which is a direct summand of soc M or S is not a composition factor of soc M ⊕ top M, then the multiplicity of S in M is one.

Let S be a simple submodule of M and consider the quotient M/S. If M/S is indecomposable, by the induction assumption it follows that M/S is multiplicityfree. Since M/S is not a projective module (since M is indecomposable), then, by theorem 6.1, it is of type A_{n−1}, where n = l(M), and so, by proposition 5.5, the multiciplity of S in M is one. If M/S is not indecomposable, then, by theorem 4.7, it follows that M/S = M_{1} ⊕ M_{2}, where each M_{i} ≠ (0) is an indecomposable nonprojective module, for i=1, 2. Again by the induction assumption and theorem 6.1, each M_{i} is of type A_{n_{i}}, where n_{i} =l(M_{i}). Hence, by proposition 5.6, the multiplicity of S in M is one. So, we conclude that each simple submodule of M has multiplicity one as a composition factor of M.

Let T be a simple module which is a composition factor of M but not a composition factor of the module soc M ⊕top M. Suppose that the multiplicity of T in M is greater than or equal to 2, that is, dim_{k}M(i_{T})≥2. Let S be a simple submodule of M. By the proof above we have that dim_{k}M(i_{S}) = 1, and so S ≅ T̸. Consider the quotient M/S. Since dim_{k}(M/S)(i_{T}) ≥ 2, by the induction assumption (for l(M/S)<l(M)) and by theorem 4.7, it follows that M/S =M_{1}⊕M_{2}, where M_{i} ≠(0), i = 1, 2, are indecomposable non-projective modules, whose dim_{k}M_{i}(i_{T}) = 1, for each i = 1, 2, and dim_{k}M(i_{T}) = 2. Moreover, since l(M_{i}) < l(M), for each i = 1, 2, by the induction assumption and theorem 6.1, each M_{i} is of type A_{n_{i}}, where n_{i} = l(M_{i}) ≥ 1. Now let π: M → M/S be the canonical epimorphism, and let the submodules N_{i} =π^{−1}M_{i}, i=1, 2, be such that M =N_{1} + N_{2}, N_{1} ∩ N_{2} =S =kv, where v ∈ M(i_{S}), and N_{i}/S = M_{i}, for i = 1, 2. From corollary 4.6, it follows that there are arrows σ_{1} : y_{1} → i_{S} and σ_{2} : y_{2} → i_{S}, where σ_{1} ≠ σ_{2} (hence, y_{1} ≠ i_{S} and y_{2} ∉ {y_{1}, i_{S}}), such that Im N_{i}(σ_{i}) = kv, for each i = 1, 2 and N_{i}(σ_{j}) = 0 if i ≠ j. Then, each N_{i} is of type A_{n_{i}+1} and i_{S} is a terminal extreme of the quiver Q_{N_{i}}, for i = 1, 2. Hence, i_{S} is a 2-sink of Q_{M} and {i_{S} , i_{T} } ⊂ supp N_{1} ∩ supp N_{2}. We claim that if z ∈supp N_{1} ∩ supp N_{2}, z ≠i_{S}, then z is a transit point of Q_{N_{1}} and of Q_{N_{2}} . In fact, since soc N_{i} ⊂ soc M, for each i = 1, 2, and each composition factor of soc M has multiplicity one in M, then the simple module S_{z} is not a composition factor of soc N_{i}, for all i=1, 2. So, z is not a sink of Q_{N_{1}} , nor of Q_{N_{2}} . On the other hand, since top M =top N_{1} ⊕top N_{2} and each composition factor of top M has multiplicity one in M (for duality of the soc M), then, for each i = 1, 2, no direct summand of top N_{i} is isomorphic to S_{z}. Therefore, z is not a source of Q_{N_{1}} , nor of Q_{N_{2}} . Our claim is proved.

Therefore, the set supp N_{1} ∩supp N_{2} contains the vertices of M which are transit point of Q_{N_{1}} and of Q_{N_{2}} . So, the neighbourhood of these vertices z ≠i_{S} in each of

these subquivers has the following figure: x_{i} →^{α_{i}} z →^{β_{i}} w_{i} such that β_{i}α_{i} ∉R_{Λ}(x_{i}, w_{i}), for each i = 1, 2. From the condition (SP)2, it follows that if α_{1} = α_{2} then β_{1} = β_{2}, and if α_{1} ≠ α_{2}, then β_{1} ≠ β_{2}, β_{2}α_{1} ∈ R_{Λ}(x_{1}, w_{2}) and β_{1}α_{2} ∈ R_{Λ}(x_{2}, w_{1}). Thus the neighbourhood of each z ∈ supp N_{1} ∩ supp N_{2}, with z ≠ i_{S}, is one of the following figures:

[figure: (a); x_{1}; z; w_{1}; α_{1}; β_{1}; with β_{1}α_{1} ∉R_{Λ}; or; (b); x_{1}; z; w_{1}; w_{2}; x_{2}; α_{1}; β_{2}; α_{2}; β_{1}; where β_{i}α_{i} ∉R_{Λ}, i=1, 2,; β_{2}α_{1} ∈R_{Λ} and β_{1}α_{2} ∈R_{_{Λ}.}]

We observe that in the case (a) we have that {x_{1}, w_{1}} ⊂ supp N_{1} ∩ supp N_{2}. For each z ∈ supp N_{1} ∩ supp N_{2}, where z ≠ i_{S}, and for each i = 1, 2, let A_{i}(i_{S}, z) be the walk of minimal length of the quiver N_{i} joining i_{S} to z. Then, for each i, A_{i}(i_{S}, z) contains the arrow σ_{i} and is, in particular, a subquiver of type A_{r_{i}}, where r_{i} = l(A_{i}(i_{S}, z)) + 1 and l(A_{i}(i_{S}, z)) denotes the number of arrows of A_{i}(i_{S}, z). Now, we choose, for each i = 1, 2, z_{i} ∈ supp N_{1} ∩ supp N_{2}, where z_{i} ≠ i_{S}, such that l(A_{i}(i_{S}, z_{i})) is the least among the vertices z ≠ i_{S} which are in supp N_{1} ∩ supp N_{2}. This choice implies that we need to annalyse two cases: z_{1} =z_{2} or z_{1} ≠z_{2}.

1\underline{st} case: z_{1} = z_{2} = z_{0}. The minimality of the length of A_{i}(i_{S}, z_{0}), i = 1, 2, implies that the neighborhood of z_{0} in Q_{M} is as in figure (b) and that i_{S} and z_{0} are the unique common vertices of A_{1}(i_{S}, z_{0}) and A_{2}(i_{S}, z_{0}). Then Q_{M} and, consequently, Q_{Λ}, contain one of the following subquivers, which are the union of the walks A_{i}(i_{S}, z_{0}), i=1, 2:

[figure: (c); z_{O}; i_{S}; y_{1}; y_{2}; σ_{2}; σ_{1}; (d); z_{O}; i_{S}; y_{1}; y_{2}; β_{2}; α_{1}; σ_{2}; σ_{1}]

where in (c) z_{0} is a 2-source or a 2-sink.

In (c), since Q_{Λ} contains no subquivers of type Ã_{r} without relations, it is easy to see that each A_{i}(i_{S}, z_{0}), i=1, 2, is a path from z_{0} to i_{S}. So, they are parallel and such that A_{1}(i_{S}, z_{0})−A_{2}(i_{S}, z_{0})∉R_{Λ} (for A_{i}(i_{S}, z_{0})α_{i} ∉R_{Λ}), which is a contradiction to condition (SP)4 of theorem 4.1. If we have case (d), we construct the families N′ =(N′(x)_{x}, N′(δ)_{δ}) and N′′ =(N′′(x)_{x}, N′′(δ)_{δ}), where x denotes the vertices of Q_{Λ} and δ their arrows, defined by:

N′(x)=\begin{cases}N_{2}(x) & if x is in A_{2}(i_{S} , z_{0}), x≠i_{S}, \\ N_{1}(x) & if x is in A_{1}(i_{S} , z_{0}), x∉\{z_{0}, i_{S} \}, \\ kv & if x=i_{S}, \\ (0) & for the other vertices, \end{cases}

N′(δ)=\begin{cases}N_{2}(δ) & if δ is in A_{2}(i_{S} , z_{0}), \\ N_{1}(δ) & if δ is in A_{1}(i_{S} , z_{0}), δ ≠α_{1}, \\ 0 & for the other arrows, \end{cases}

N′′(x)=\begin{cases}N_{2}(x) & if x is in A_{2}(i_{S} , z_{0}), x∉\{z_{0}, i_{S} \}, \\ N_{1}(x) & if x is in A_{1}(i_{S} , z_{0}), x≠i_{S}, \\ kv & if x=i_{S}, \\ (0) & for the other vertices, \end{cases}

N′′(δ)=\begin{cases}N_{2}(δ) & if δ is in A_{2}(i_{S} , z_{0}), δ ≠β_{2}, \\ N_{1}(δ) & if δ is in A_{1}(i_{S} , z_{0}), \\ 0 & for the other arrows. \end{cases}

By the definition of N′ and N′′, it is easy to verify that they are k-representations such that supp N′ = supp N′′. Moreover, the quivers Q_{N′} and Q_{N′′} are obtained from the quiver in (d) by deleting, respectively, the arrows α_{1} and β_{2}; and it follows that N′ (N′′) is a module of type A_{r}, where r =l(A_{1}(i_{S}, z_{0})) + l(A_{2}(i_{S}, z_{0})), whose top (respectively, whose soc) contains a direct summand isomorphic to the simple module S_{z_{0}} , since the vertex z_{0} is a initial extreme of Q_{N′} (respectively, since z_{0} is a terminal extreme of Q_{N′′}). So we construct two multiplicity-free modules (N′ and N′′) having the same composition factors, but non-isomorphic, which is a contradiction.

2\underline{nd} case: z_{1} ≠ z_{2}. From the minimality of the length of A_{1}(i_{S}, z_{1}) and of A_{2}(i_{S}, z_{2}), it follows that i_{S} is the unique common vertex of them and, analogously to the first case, the neighbourhood of z_{1} and of z_{2} in Q_{M} are as in figure (b). So, we indicate their arrows as well as the vertices of the neighbourhood of each z_{i}, using double indices, in the following way: the first index i corresponds to the vertex z_{i} and the second one j to which Q_{N_{j}} belongs, as the following figure shows:

[figure: x_{i,1}; z_{i}; w_{i,1}; w_{i,2}; x_{i,2}; α_{i,1}; β_{i,2}; α_{i,2}; β_{i,1}; with β_{i,1}α_{i,1} ∉R_{Λ}, β_{i,2}α_{i,2} ∉R_{Λ}, β_{i,2}α_{i,1} ∈R_{Λ}; and β_{i,1}α_{i,2} ∈R_{Λ}, for i=1, 2.]

Since, for each i=1, 2, Q_{N_{i}} is connected, we consider the walk A_{1}(z_{1}, z_{2}) in Q_{N_{1}} and the walk A_{2}(z_{1}, z_{2}) in Q_{N_{2}} , such that they are the walks of the least length, in each corresponding quiver, joining z_{1} to z_{2}. It is clear, by the assumption on minimality of the length, that z_{1} and z_{2} are their unique common vertices. So, we have many different possibilities, according to whether the arrows α_{i,j} and β_{i,j}, for i, j = 1, 2, belong (or not) to the walks A_{i}(i_{S}, z_{i}) and A_{i}(z_{1}, z_{2}), which are the following.

2.1: For all i = 1, 2, A_{i}(i_{S}, z_{i}) contains the arrow β_{i,i}. Under this condition, each walk A_{i}(z_{1}, z_{2}) contains the arrow α_{i,i} and exactly one of α_{i,j} or β_{i,j} if i ≠ j. Therefore, there are two subcases.

2.1.1: For some i, for example i = 1, A_{1}(z_{1}, z_{2}) contains the arrow β_{2,1}. Then the union of the walks A_{i}(i_{S}, z_{i}), for i=1, 2 and A_{1}(z_{1}, z_{2}) determines a subquiver of type Ã_{r} without relations, where r is the sum of the lengths of those walks, which is illustrated by

[figure: z_{2}; z_{1}; β_{2,1}; α_{1,1}; σ_{2}; σ_{1}; β_{2,2}; β_{1,1}; i_{S}; where β_{1,1}α_{1,1} ∉R_{Λ}.]

This contradicts the fact that Λ is a representation-finite algebra.

2.1.2: For some i, for example i=1, A_{1}(z_{1}, z_{2}) contains the arrow α_{2,1}. Under this condition, Q_{M} contains a subquiver which is the union of the walks A_{i}(i_{S}, z_{i}), i=1, 2, and A_{1}(z_{1}, z_{2}), and it is illustrated by

[figure: z_{2}; z_{1}; α_{2,1}; α_{1,1}; σ_{2}; σ_{1}; β_{2,2}; β_{1,1}; i_{S}; where β_{1,1}α_{1,1} ∉R_{Λ} and; β_{2,2}α_{2,1} ∈R_{Λ}.]

In this subcase, let N′ = (N′(x)_{x}, N′(α)_{α}) and N′′ = (N′′(x)_{x}, N′′(α)_{α}) be the families defined by

N′(x)=\begin{cases}N_{2}(x) & if x∈A_{2}(i_{S} , z_{2}), x∉\{z_{2}, i_{S} \}, \\ N_{1}(x) & if x∈A_{1}(i_{S} , z_{1}) ∪ A_{1}(z_{1}, z_{2}) and x≠i_{S}, \\ kv & if x=i_{S}, \\ (0) & for the other vertices, \end{cases}

N′(α)=\begin{cases}N_{2}(α) & if α∈A_{2}(i_{S} , z_{2}) and α≠β_{2,2}, \\ N_{1}(α) & if α∈A_{1}(i_{S} , z_{1}) ∪ A_{1}(z_{1}, z_{2}), \\ 0 & for the other arrows, \end{cases}

N′′(x)=\begin{cases}N_{2}(x) & if x∈A_{2}(i_{S} , z_{2}) and x≠i_{S}, \\ N_{1}(x) & if x∈A_{1}(i_{S} , z_{1}) ∪ A_{1}(z_{1}, z_{2}) and x∉\{z_{2}, i_{S} \}, \\ kv & if x=i_{S}, \\ (0) & for the other vertices, \end{cases}

N′′(α)=\begin{cases}N_{2}(α) & if α∈A_{2}(i_{S} , z_{2}), \\ N_{1}(α) & if α∈A_{1}(i_{S} , z_{1}) ∪ A_{1}(z_{1}, z_{2}), \\ & and α≠α_{2,1}, \\ 0 & for the other arrows. \end{cases}

It is easy to verify that N′ and N′′ are k-representations whose quivers are obtained from the quiver above by deleting the arrows β_{2,2} and α_{2,1}, respectively. Therefore, using the same arguments used in 2.1.2, we conclude that N′ and N′′ are indecomposable multiplicity-free modules having the same composition factors (since they have the same support), but they are not isomorphic (since S_{z_{2}} ⊂ soc N′ and S_{z_{2}} ⊄ soc N′′). Again we obtain a contradiction to the assumption on Λ-ind.

2.2: For some i, for example i = 1, A_{1}(i_{S}, z_{1}) contains the arrow α_{1,1}. Then, A_{1}(z_{1}, z_{2}) necessarily contains the arrow β_{1,1}. Relative to the other arrows to which z_{1} and z_{2} belong, we have the following subcases.

2.2.1: A_{1}(z_{1}, z_{2}) contains the arrow β_{2,1} (or α_{2,1}) and A_{2}(i_{S}, z_{2}) contains the arrow β_{2,2} (respectively, α_{2,2});

2.2.2: A_{1}(z_{1}, z_{2}) contains β_{2,1} and A_{2}(i_{S}, z_{2}) contains α_{2,2}; and 2.2.3: A_{1}(z_{1}, z_{2}) contains α_{2,1} and A_{2}(i_{S}, z_{2}) contains β_{2,2}. Since subcase 2.2.1

is similar to subcase 2.1.1 and the others are similar to 2.1.2, we leave it to the reader to verify that they lead to a contradiction to the assumptions. Hence, we cannot suppose that T does not have multiplicity one as a composition factor of M. So it is proved that each indecomposable module is multiplicity-free.

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Departamento de Matemática-IMEUSP, Universidade de São Paulo, CP 66281 - CEP 05315-970, São Paulo, Brazil

E-mail address: bel@ime.usp.br