PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 5, Pages 1313–1322 S 0002-9939(99)04714-0 Article electronically published on January 27, 1999

DEGENERATIONS FOR MODULES OVER REPRESENTATION-FINITE ALGEBRAS

GRZEGORZ ZWARA

(Communicated by Ken Goodearl)

Abstract. Let A be a representation-finite algebra. We show that a finite dimensional A-module M degenerates to another A-module N if and only if the inequalities dim_{K} Hom_{A}(M, X) ≤ dim_{K} Hom_{A}(N, X) hold for all A-modules X. We prove also that if Ext_{A}^{1}(X, X) = 0 for any indecomposable A-module X, then any degeneration of A-modules is given by a chain of short exact sequences.

1. Introduction and main results

Let A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K. If a_{1} = 1, …, a_{n} is a basis of A over K, we have the structure constants a_{ijk} defined by a_{i}a_{j} = ∑ a_{ijk}a_{k}. The affine variety mod_{A}(d) of d-dimensional unital left A-modules consists of n-tuples m = (m_{1}, …, m_{n}) of d×dmatrices with coefficients in K such that m_{1} is the identity matrix and m_{i}m_{j} = ∑ a_{ijk}m_{k} holds for all indices i and j. The general linear group Gl_{d}(K) acts on mod_{A}(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules (see [11]). We shall agree to identify a d-dimensional Amodule M with the point of mod_{A}(d) corresponding to it. We denote by O(M) the Gl_{d}(K)-orbit of a module M in mod_{A}(d). Then one says that a module N in mod_{A}(d) is a degeneration of a module M in mod_{A}(d) if N belongs to the Zariski closure \overline{O(M)} of O(M) in mod_{A}(d), and we denote this fact by M ≤_{deg} N. Thus ≤_{deg} is a partial order on the set of isomorphism classes of A-modules of a given dimension. It is not clear how to characterize ≤_{deg} in terms of representation theory.

There has been a work by S. Abeasis and A. del Fra [1], K. Bongartz [6], [9], [8], Ch. Riedtmann [13], and A. Skowroński and the author [15], [16], [17] and [18] connecting ≤_{deg} with other partial orders ≤_{ext} and ≤ on the isomorphism classes in mod_{A}(d). They are defined in terms of representation theory as follows:

• M ≤_{ext} N: ⇔ there are modules M_{i}, U_{i}, V_{i} and short exact sequences 0 → U_{i} → M_{i} → V_{i} → 0 in mod A such that M = M_{1}, M_{i+1} = U_{i} ⊕ V_{i}, 1 ≤ i ≤ s, and N = M_{s+1} for some natural number s.

• M ≤ N: ⇔ [M, X] ≤ [N, X] holds for all modules X.

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors May 6, 1997 and, in revised form, August 28, 1997.

1991 Mathematics Subject Classification. Primary 14L30, 16G60, 16G70.

Here and later on we abbreviate dim_{K} Hom_{A}(X, Y) by [X, Y]. Then for modules M and N in mod_{A}(d), the following implications hold:

M ≤_{ext} N ⟹ M ≤_{deg} N ⟹ M ≤ N

(see [9], [13]). Unfortunately, the reverse implications are not true in general, and it would be interesting to find out when they are. K. Bongartz proved in [9] that it is the case for all representations of Dynkin quivers and the double arrow. Moreover, in [8] K. Bongartz proved that ≤_{deg} and ≤ coincide for all modules over tame concealed algebras. Recently, the author proved in [17] that ≤ and ≤_{ext} are also equivalent for all modules over representation-finite blocks of group algebras, and in [18] that ≤_{ext} and ≤_{deg} coincide for all modules over tame concealed algebras. The main aim of this paper is to prove the following theorem.

Theorem 1. Let A be a representation-finite algebra and M, N two modules with M ≤ N. Then there are A-modules Z, Z′, and two exact sequences

0 → N → M ⊕ Z → Z → 0 and 0 → Z′ → M ⊕ Z′ → N → 0.

In [13] Riedtmann proved that each of the exact sequences 0 → N → M ⊕ Z → Z → 0 and 0 → Z′ → M ⊕ Z′ → N → 0 implies that M ≤_{deg} N. Hence we get the following fact which solves a long standing problem (see [13]).

Corollary. The partial orders ≤ and ≤_{deg} coincide for all modules over representation-finite algebras.

We note that for a representation-finite algebra A we may deduce the dimension of the spaces Hom_{A}(M, N) from the Auslander-Reiten quiver of A (see [10]), and hence it is rather easy to decide when M ≤ N for any A-modules M and N.

There are many examples of representation-finite algebras for which the orders ≤_{deg} and ≤_{ext} are not equivalent (see [17]). Our second aim in this paper is to prove the following theorem.

Theorem 2. Let B be an algebra and assume that Ext_{B}^{1}(X, X) = 0 for any indecomposable B-module X. Then the partial orders ≤, ≤_{deg} and ≤_{ext} coincide for all B-modules.

It is well-known that every representation-directed algebra [14] satisfies the above condition. Hence, Theorem 2 extends the corresponding result by Bongartz proved in [9].

The paper is organized as follows. In Section 2 we fix the notation, recall the relevant definitions and facts, and prove some preliminary results on modules which we apply in our investigations. Section 3 is devoted to the proofs of Theorems 1 and 2.

For basic background on the topics considered here we refer to [5], [9], [8], [11] and [14]. The results presented in this paper form a part of the author’s doctoral dissertation written under the supervision of Professor A. Skowroński. The author gratefully acknowledges support from the Polish Scientific Grant KBN No. 2 PO3A 020 08.

2. Preliminary results

2.1. Throughout the paper A denotes a fixed finite dimensional associative Kalgebra with an identity over an algebraically closed field K. We denote by mod A the category of finite dimensional left A-modules and by rad(mod A) the Jacobson radical of mod A. By an A-module we mean an object from mod A. Further, we denote by Γ_{A} the Auslander-Reiten quiver of A, and by τ = τ_{A} and τ^{−} = τ_{A}^{−} the Auslander-Reiten translations D Tr and Tr D, respectively. We shall agree to identify the vertices of Γ_{A} with the corresponding indecomposable modules. For a module M we denote by [M] the image of M in the Grothendieck group K_{0}(A) of A. Thus [M] = [N] if and only if M and N have the same simple composition factors including the multiplicities.

2.2. Following [13], for M, N from mod A, we set M ≤ N if and only if [M, X] ≤ [N, X] for all A-modules X. The fact that ≤ is a partial order on the isomorphism classes of A-modules follows from a result by M. Auslander [3] (see also [6]). Observe that, if M and N have the same dimension and M ≤ N, then [M] = [N]. Moreover, M. Auslander and I. Reiten have shown in [4] that, if M and N are A-modules with [M] = [N], then for all nonprojective indecomposable A-modules X and all noninjective indecomposable modules Y the following formulas hold (see [12]):

[X, M] − [M, τX] = [X, N] − [N, τX],
[M, Y] − [τ^{−}Y, M] = [N, Y] − [τ^{−}Y, N].

Hence, if [M] = [N], then M ≤ N if and only if [X, M] ≤ [X, N] for all A-modu-les X.

2.3. Let M and N be A-modules with [M] = [N] and

Σ : 0 → D → E → F → 0

an exact sequence in mod A. Following [13] we define the additive functions δ_{M,N}, δ′_{M,N} and δ_{Σ} on A-modules X as follows:

δ_{M,N}(X) = [N, X] − [M, X],
δ′_{M,N}(X) = [X, N] − [X, M],
δ_{Σ}(X) = δ_{E,D⊕F} (X) = [D ⊕ F, X] − [E, X],
δ′_{Σ}(X) = δ′_{E,D⊕F} (X) = [X, D ⊕ F] − [X, E].

From the Auslander-Reiten formulas (2.2) we get the following very useful equalities:

δ_{M,N}(X) = δ′_{M,N}(τ^{−}X), δ_{M,N}(τX) = δ′_{M,N}(X)

for all A-modules X. Observe also that δ_{M,N}(I) = 0 for any injective A-module I, and δ′_{M,N}(P) = 0 for any projective A-module P. In particular, the following conditions are equivalent:

(1) M ≤ N,

(2) δ_{M,N}(X) ≥ 0 for all X ∈ Γ_{A},

(3) δ′_{M,N}(X) ≥ 0 for all X ∈ Γ_{A}.

2.4. For an A-module M and an indecomposable A-module Z, we denote by μ(M, Z) the multiplicity of Z as a direct summand of M. For a noninjective indecomposable A-module U, we denote by Σ(U) an Auslander-Reiten sequence

Σ(U) : 0 → U → E(U) → τ^{−}U → 0.

We shall need the following lemma.

Lemma 2.5. Let M, N be A-modules with [M] = [N] and U an indecomposable A-module. Then:

(i) If U is noninjective, then δ_{Σ(U)}(M) = μ(M, U). (ii) If M ≤ N, then μ(N, U) − μ(M, U) ≤ δ_{M,N}(U) + δ′_{M,N}(U).

Proof. If U is noninjective, then the Auslander-Reiten sequence Σ(U) induces an exact sequence

0 → Hom_{A}(τ^{−}U, M) → Hom_{A}(E(U), M) → rad(U, M) → 0,

and hence we get

[U ⊕ τ^{−}U, M] − [E(U), M] = [U, M] − dim_{K} rad(U, M) = μ(M, U).

This implies (i). Similarly, we have

[U ⊕ τ^{−}U, N] − [E(U), N] = μ(N, U).

Then we obtain

μ(N, U) − μ(M, U) = ([U ⊕ τ^{−}U, N] − [U ⊕ τ^{−}U, M]) − ([E(U), N] − [E(U), M])
= δ′_{M,N}(U) + δ′_{M,N}(τ^{−}U) − δ′_{M,N}(E(U))
≤ δ′_{M,N}(U) + δ′_{M,N}(τ^{−}U) = δ′_{M,N}(U) + δ_{M,N}(U).

Assume now that U is injective. Then Hom_{A}(U/ soc(U), M) ≃ rad(U, M), and so

[U, M] − [U/ soc(U), M] = μ(M, U).

Similarly, we have

[U, N] − [U/ soc(U), N] = μ(N, U).

Therefore, we get

μ(N, U) − μ(M, U) = ([U, N] − [U, M]) − ([U/ soc(U), N] − [U/ soc(U), M])
= δ′_{M,N}(U) − δ′_{M,N}(U/ soc(U)) ≤ δ′_{M,N}(U)
= δ′_{M,N}(U) + δ_{M,N}(U).

Hence, (ii) also holds.

We shall need also the following Lemma (3 + 3 + 2) from [2, (2.1)] and its direct consequence.

Lemma 2.6. Let

\begin{matrix}Σ_{1} : 0 → M_{1} & \underrightarrow{[ \begin{matrix}u_{1} \\ f_{1} \end{matrix} ]} & M_{2} ⊕ N_{1} & ⟶^{[f_{2},u_{2}]} & N_{2} → 0, \\ Σ_{2} : 0 → M_{2} & \underrightarrow{[ \begin{matrix}v_{1} \\ f_{2} \end{matrix} ]} & M_{3} ⊕ N_{2} & ⟶^{[f_{3},v_{2}]} & N_{3} → 0 \end{matrix}

be short exact sequences in mod A. Then the sequence

Σ_{3} : 0 → M_{1} \underrightarrow{[ \begin{matrix}v_{1}u_{1} \\ f_{1} \end{matrix} ]} M_{3} ⊕ N_{1} ⟶^{[f_{3}, −v_{2}u_{2} ]} N_{3} → 0

is exact. Moreover, we have δ_{Σ_{3}} = δ_{Σ_{1}} + δ_{Σ_{2}}.

2.7. A short exact sequence

0 → U ⟶^{f} W ⟶^{g} V → 0

is said to be without isomorphism provided f ∈ rad(U, W) and g ∈ rad(W, V). Let Σ : 0 → U ⟶^{f} W ⟶^{g} V → 0 be any exact sequence. It is easy to see that if f ∈ rad(U, W), then there is an exact sequence without isomorphism 0 → U → W′ → V′ → 0 such that W = W′ ⊕ Y and V = V′ ⊕ Y for some A-modules W′, V′ and Y. Dually, if g ∈ rad(W, V), then there is an exact sequence without isomorphism 0 → U′ → W′ → V → 0 such that U = U′ ⊕ Z and W = W′ ⊕ Z for some A-modules U′, W′ and Z. Moreover, if Σ is nonsplittable, then there is a nonsplittable exact sequence without isomorphism 0 → U′ → W′ → V′ → 0 such that U = U′ ⊕ Y, W = W′ ⊕ Y ⊕ Z and V = V′ ⊕ Z for some A-modules U′, W′, V′, Y and Z.

Lemma 2.8. Let Σ : 0 → U ⟶^{f} W ⟶^{g} V → 0 be an exact sequence without isomorphism. Then:

(i) For any nonzero direct summand U′ of U, δ_{Σ}(U′) > 0 holds. (ii) For any nonzero direct summand V′ of V, δ′_{Σ}(V′) > 0 holds.

Proof. (i) Let U′ be a nonzero direct summand of U. The sequence Σ induces an exact sequence

0 → Hom_{A}(V, U′) → Hom_{A}(W, U′) ⟶^{f^{∗}} Hom_{A}(U, U′).

Assume that f^{∗} is an epimorphism. Then there is a homomorphism of A-modules h : W → U′ such that f^{∗}(h) = hf : U → U′ is a projection. But then f ∉ rad(U, W), which yields a contradiction. Hence, [V, U′] − [W, U′] + [U, U′] > 0, and consequently δ_{Σ}(U′) > 0.

The proof of (ii) is dual.

As a consequence of the above lemma, we get the following fact.

Lemma 2.9. Let Σ : 0 → U → W → V → 0 be a nonsplittable exact sequence. Then δ_{Σ}(U) > 0 and δ′_{Σ}(V) > 0.

Lemma 2.10. Let X be an A-module and Σ : 0 → U → W → V → 0 a nonsplittable short exact sequence of A-modules.

(i) If δ_{Σ}(X) > 0, then there exists a nonsplittable exact sequence of A-modules

Φ : 0 → X → Y → V → 0,

such that δ_{Φ} ≤ δ_{Σ}. (ii) If δ′_{Σ}(X) > 0, then there exists a nonsplittable exact sequence of A-modules

Φ : 0 → U → Z → X → 0,

such that δ_{Φ} ≤ δ_{Σ}.

Proof. (i) The first part of the proof is due to U. Markolf (see the proof of Theorem 4 in [7]). Let X be an A-module such that δ_{Σ}(X) > 0. Then the last map in the following exact sequence

0 → Hom_{A}(V, X) → Hom_{A}(W, X) → Hom_{A}(U, X) → Ext_{A}^{1}(V, X) → Ext_{A}^{1}(W, X)

is not a monomorphism. Therefore, we find a nonsplittable exact sequence of Amodules Φ : 0 → X → Y → V → 0, whose pullback under W → V is a splittable sequence. Thus we get the following commutative diagram with exact rows and columns:

0  0
↓  ↓
U  = U
↓  ↓
0 → X ⟶ X ⊕ W ⟶ W → 0
||  ↓  ↓
0 → X ⟶  Y  ⟶ V → 0
↓  ↓
0  0

So, we have an exact sequence Θ : 0 → U → X ⊕ W → Y → 0. Observe that δ_{Σ} = δ_{Φ} + δ_{Θ}. This implies that δ_{Φ} ≤ δ_{Σ}.

The proof of (ii) is dual.

Lemma 2.11. If M <_{deg} N, then δ_{M,N}(N) > 0 and δ′_{M,N}(N) > 0.

Proof. Suppose that δ′_{M,N}(N) = 0. By Theorem 2.4 in [9], we know that if a module U embeds into N and [U, N] = [U, M], then U also embeds into M. Applying this fact for U = N, we obtain that N embeds into M. But the modules M and N have the same dimension. This implies that M is isomorphic to N, which gives a contradiction. Hence, δ′_{M,N}(N) > 0 and δ_{M,N}(N) > 0 by duality.

3. Proof of Theorems 1 and 2

Throughout this section A denotes a representation-finite algebra.

Lemma 3.1. Let M and N be two A-modules with M < N, and let

Σ : 0 → U → W → V → 0

be a short exact sequence without isomorphism in mod A such that δ_{Σ} ≤ δ_{M,N}. Then there exists a short exact sequence without isomorphism in mod A

Φ : 0 → U → Y → Z → 0

such that δ_{Σ} ≤ δ_{Φ} ≤ δ_{M,N} and δ_{Φ}(Y) = δ_{M,N}(Y).

Proof. Let

Σ : 0 → U → W → V → 0

be a short exact sequence of A-modules without isomorphism such that δ_{Σ} ≤ δ_{M,N}. Take a short exact sequence without isomorphism in mod A,

Φ : 0 → U → Y → Z → 0

such that δ_{Σ} ≤ δ_{Φ} ≤ δ_{M,N}, and which is maximal in the following sense. For any short exact sequence without isomorphism Φ′ in mod A starting at U and satisfying inequalities δ_{Φ} ≤ δ_{Φ′} ≤ δ_{M,N}, we have δ_{Φ} = δ_{Φ′}. Since ∑_{X∈Γ_{A}} δ_{M,N}(X) is finite, such a sequence Φ exists. Assume now that Y = Y_{1}⊕Y_{2}, where Y_{1} is indecomposable with δ_{Φ}(Y_{1}) < δ_{M,N}(Y_{1}). Then Y_{1} is noninjective and we have an Auslander-Reiten sequence

Σ(Y_{1}) : 0 → Y_{1} ⟶^{h} E → τ^{−}Y_{1} → 0,

and of course

Φ : 0 → U → Y_{1} ⊕ Y_{2} ⟶^{(f_{1},f_{2})} Z → 0.

Since f_{1} ∈ rad(Y_{1}, Z), the push out of the Auslander-Reiten sequence Σ(Y_{1}) is a splittable sequence, so we obtain the following commutative diagram with exact rows:

0 → Y_{1} ⟶^{h}  E  ⟶ τ^{−}Y_{1} → 0
↓ ^{f_{1}}  ↓  ||
0 → Z ⟶ τ^{−}Y_{1} ⊕ Z ⟶ τ^{−}Y_{1} → 0.

This implies that there exists a nonsplittable exact sequence

Ψ : 0 → Y_{1} \underrightarrow{(\begin{matrix}h \\ f_{1} \end{matrix})} E ⊕ Z → τ^{−}Y_{1} ⊕ Z → 0.

Applying Lemma 2.6 for Φ and Ψ, we get a new exact sequence

0 → U ⟶^{ı} Y_{2} ⊕ E → Z ⊕ τ^{−}Y_{1} → 0.

Since Φ is a sequence without isomorphism, we have ı ∈ rad(U, Y_{2} ⊕ E). Hence, there is a sequence without isomorphism in mod A

Θ : 0 → U → \overline{Y} → \overline{Z} → 0,

with Y_{2} ⊕ E = \overline{Y} ⊕ \overline{W} and Z ⊕ τ^{−}Y_{1} = \overline{Z} ⊕ \overline{W} for some A-module \overline{W}. Thus, by Lemmas 2.6 and 2.5(i), for any A-module X we have

δ_{Θ}(X) = δ_{Φ}(X) + δ_{Ψ}(X) = δ_{Φ}(X) + δ_{Σ(Y_{1})}(X) = δ_{Φ}(X) + μ(X, Y_{1}).

Since δ_{Φ} ≤ δ_{M,N} and δ_{Φ}(Y_{1}) ≤ δ_{M,N}(Y_{1}) − 1, we get δ_{Σ} ≤ δ_{Θ} ≤ δ_{M,N}. This gives a contradiction with our choice of the sequence Φ. Hence, δ_{Φ}(Y) = δ_{M,N}(Y), and this finishes the proof.

Lemma 3.2. If M < N, then δ_{M,N}(N) > 0 and δ′_{M,N}(N) > 0.

Proof. We proceed by induction on ∑_{X∈Γ_{A}} δ_{M,N}(X) > 0. Applying equalities (2.3), we obtain ∑_{X∈Γ_{A}} δ_{M,N}(X) = ∑_{X∈Γ_{A}} δ′_{M,N}(X). Assume M < N and that δ_{M,N}(N) = 0 or δ′_{M,N}(N) = 0. By duality, we may assume that δ′_{M,N}(N) = 0 and moreover, the modules M and N have no nonzero common direct summand. Let F be the set of all modules in Γ_{A} which are a direct summands of N. Take Y ∈ F. By Lemma 2.5(ii), we get

μ(N, Y) = μ(N, Y) − μ(M, Y) ≤ δ_{M,N}(Y) + δ′_{M,N}(Y) = δ_{M,N}(Y).

So, the module Y is noninjective and there is an Auslander-Reiten sequence Σ(Y). We define a new exact sequence without isomorphism

Σ : 0 → N → E(N) → τ^{−}N → 0,

where E(N) = ⨁_{Y ∈F} E(Y)^{μ(N,Y)} and τ^{−}N = ⨁_{Y ∈F}(τ^{−}Y)^{μ(N,Y)}. Applying Lemma 2.5(i), we obtain

δ_{Σ}(Y) = μ(N, Y) ≤ δ_{M,N}(Y),

for any Y ∈ Γ_{A}. Consequently δ_{Σ} ≤ δ_{M,N} and, from Lemma 3.1, there is an exact sequence without isomorphism

Φ : 0 → N → W → V → 0

with δ_{Φ} ≤ δ_{M,N} and δ_{Φ}(W) = δ_{M,N}(W). Then M ⊕ V ≤ W and δ_{M⊕V,W} (W) = 0. Observe that δ_{M,N} − δ_{M⊕V,W} = δ_{Φ} and, from Lemma 2.9, δ_{Φ}(N) > 0. This leads to

∑_{X∈Γ_{A}} δ_{M⊕V,W} (X) < ∑_{X∈Γ_{A}} δ_{M,N}(X).

It follows from our inductive assumption that the modules M ⊕ V and W are isomorphic. Then the sequence Φ has the form

0 → N → V ⊕ M → V → 0,

and this implies that M <_{deg} N, by Proposition 3.4 in [13]. Applying Lemma 2.11, we get δ′_{M,N}(N) > 0, and hence a contradiction. This finishes the proof.

3.3. Proof of Theorem 1. Let M and N be A-modules with M ≤ N. We may assume that M < N. Let r(X) = min{δ_{M,N}(X), μ(N, X)}, for any X ∈ Γ_{A}, and let F be the set of all vertices of Γ_{A} with r(X) > 0. The set F does not contain injective A-modules and is nonempty, by Lemma 3.2. Let N′ = ⨁_{X∈F} X^{r(X)} = ⨁_{X∈Γ_{A}} X^{r(X)} and N′′ = ⨁_{X∈Γ_{A}} X^{μ(N,X)−r(X)}. Then N = N′ ⊕ N′′. We define a new exact sequence without isomorphism

Σ : 0 → ⨁_{X∈F} X^{r(X)} → ⨁_{X∈F} E(X)^{r(X)} → ⨁_{X∈F}(τ^{−}X)^{r(X)} → 0.

Applying Lemma 2.5(i), we obtain δ_{Σ}(X) = r(X) ≤ δ_{M,N}(X), for any X ∈ Γ_{A}. Consequently, δ_{Σ} ≤ δ_{M,N} and, by Lemma 3.1, there is an exact sequence without isomorphism

Φ : 0 → N′ → W → Z → 0

with δ_{Σ} ≤ δ_{Φ} ≤ δ_{M,N} and δ_{Φ}(W) = δ_{M,N}(W). Then M ⊕ Z ≤ N′′ ⊕ W and δ_{M⊕Z,N′′⊕W} (W) = 0. Let N_{1} be any indecomposable direct summand of N′′. Then r(N_{1}) < μ(N, N_{1}), and this leads to δ_{Σ}(N_{1}) = r(N_{1}) = δ_{M,N}(N_{1}). Hence,

δ_{M⊕Z,N′′⊕W} (N_{1}) = δ_{M,N}(N_{1}) − δ_{Φ}(N_{1}) = δ_{Σ}(N_{1}) − δ_{Φ}(N_{1}) ≤ 0.

So, δ_{M⊕Z,N′′⊕W} (N_{1}) = 0. This implies that δ_{M⊕Z,N′′⊕W} (N′′) = 0, and furthermore δ_{M⊕Z,N′′⊕W} (N′′ ⊕W) = 0. Hence, M ⊕Z ≃ N′′ ⊕W, by Lemma 3.2. Finally, the sequence Φ induces an exact sequence 0 → N′ ⊕ N′′ → N′′ ⊕ W → Z → 0, which has the form 0 → N → M ⊕ Z → Z → 0. In a similar way we obtain an exact sequence 0 → Z′ → M ⊕ Z′ → N → 0.

Lemma 3.4. Let M, N and X be A-modules such that M < N and X ∈ Γ_{A}. Then we have:

(i) If δ′_{M,N}(X) > 0, then there exist an indecomposable direct summand N_{1} of N and a nonsplittable exact sequence Φ : 0 → N_{1} → Y → X → 0 without isomorphism such that δ_{Φ} ≤ δ_{M,N}.

(ii) If δ_{M,N}(X) > 0, then there exist an indecomposable direct summand N_{1} of N and a nonsplittable exact sequence Φ : 0 → X → Y → N_{1} → 0 without isomorphism such that δ_{Φ} ≤ δ_{M,N}.

Proof. (i) Assume that δ′_{M,N}(X) > 0. Applying Theorem 1 we get the exact sequence Σ : 0 → N → M ⊕ Z → Z → 0, in mod A. Further, applying Lemma 2.10(ii), we obtain a nonsplittable exact sequence Ψ : 0 → N → W → X → 0 with δ_{Ψ} ≤ δ_{Σ} = δ_{M,N}. Then, by Lemma 2.9, there is an indecomposable direct summand N_{1} of N with δ_{Ψ}(N_{1}) > 0. Finally, by Lemma 2.10(i), we obtain a nonsplittable exact sequence Φ : 0 → N_{1} → Y → X → 0 with δ_{Φ} ≤ δ_{Ψ} ≤ δ_{M,N}.

We obtain (ii) by duality.

3.5. Proof of Theorem 2. Let B be an algebra and assume that Ext_{B}^{1}(X, X) = 0 for any indecomposable B-module X. It is well-known that then B is representationfinite. Let M and N be two B-modules with M ≤ N. We shall show that M ≤_{ext} N. We proceed by induction on [N, N] − [M, M] ≥ 0. If [N, N] − [M, M] = 0, then by Lemma 1.2 in [9], M is isomorphic to N. Hence, we may assume that M < N, and that M and N have no common nonzero direct summand. Take any indecomposable direct summand N_{1} of N. Applying Lemma 2.5(ii), we obtain that δ_{M,N}(N_{1}) + δ′_{M,N}(N_{1}) > 0. Without loss of generality, we may assume that δ_{M,N}(N_{1}) > 0. Now applying Lemma 3.4, we get a nonsplittable exact sequence

Σ : 0 → N_{1} → Y → N_{2} → 0

with δ_{Σ} ≤ δ_{M,N}, for some A-module Y and some indecomposable direct summand N_{2} of N. Since Ext_{B}^{1}(N_{1}, N_{1}) = 0, the modules N_{1} and N_{2} are not isomorphic. Thus, N = N_{1} ⊕ N_{2} ⊕ N_{3}, for some A-module N_{3}. Moreover, M ≤ Y ⊕ N_{3} <_{ext} N. This implies that [Y ⊕ N_{3}, Y ⊕ N_{3}] < [N, N], by Lemma 1.2 in [9]. Then

[Y ⊕ N_{3}, Y ⊕ N_{3}] − [M, M] < [N, N] − [M, M]

and M ≤_{ext} Y ⊕ N_{3}, by our inductive assumption. Finally, we obtain M <_{ext} N, and this finishes the proof.

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Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

   E-mail address: gzwara@mat.uni.torun.pl