TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 9, Pages 3857–3868 S 0002-9947(99)02137-6 Article electronically published on April 20, 1999

ON THE NUMBER OF TERMS IN THE MIDDLE OF ALMOST SPLIT SEQUENCES OVER TAME ALGEBRAS

J. A. DE LA PEÑA AND M. TAKANE

Abstract. Let A be a finite dimensional tame algebra over an algebraically closed field k. It has been conjectured that any almost split sequence 0 → X → ⊕_{i=1}^{n}Y_{i} → Z → 0 with Y_{i} indecomposable modules has n ≤ 5 and in case n = 5, then exactly one of the Y_{i} is a projective-injective module. In this work we show this conjecture in case all the Y_{i} are directing modules, that is, there are no cycles of non-zero, non-iso maps Y_{i} = M_{1} → M_{2} → ⋯ → M_{s} = Y_{i} between indecomposable A-modules. In case, Y_{1} and Y_{2} are isomorphic, we show that n ≤ 3 and give precise information on the structure of A.

Let A be a finite dimensional algebra over an algebraically closed field k. We denote by mod_{A} the category of finite dimensional left A-modules (an object in mod_{A} is simply called a module).

For a non-projective indecomposable module X, there exist an indecomposable non-injective module τ_{A}X called the Auslander-Reiten translate and an almost split sequence 0 → τ_{A}X → E → X → 0 (see [2], [18]). Since their introduction, almost split sequences have played a central role in the representation theory of algebras (see for example [2]).

For an almost split sequence 0 → τ_{A}X → E → X → 0, consider the indecom-

posable decomposition E = ⨁_{i=1}^{s(X)} Y_{i}. There has been considerable attention paid to

the relation between properties of the algebra A and the values s(X) for different modules X (and of course between properties of X and the value s(X)). Among other interesting results we recall that if A is representation finite, then s(X) ≤ 4, for every indecomposable non-projective module X [3], [7] (see also [10] and [12]).

It has been conjectured by S. Brenner that, for A a tame algebra, s(X) ≤ 5 for every indecomposable non-projective module X. This is known to hold for many examples, in particular for the important case of hereditary tame algebras.

To state the main results of this work, we recall some concepts. A cycle in mod_{A} is a sequence X_{0} ⟶^{f_{1}} X_{1} ⟶^{f_{2}} ⋯ ⟶^{f_{s}} X_{s} = X_{0} of non-zero nonisomorphism maps between indecomposable modules; the cycle is said to be finite if f_{i} ∉ rad_{A}^{∞}(X_{i−1}, X_{i}), for all i = 1, …, s. An indecomposable module X is said to be directing if it does not belong to a cycle in mod_{A}. The algebra A is said to be cycle-finite if all cycles in mod_{A} are finite. We recall that a cycle finite algebra is tame [1].

The Coxeter matrix ϕ_{A} of A and its spectral radius ρ(ϕ_{A}) = max {∥λ∥: λ eigenvalue of ϕ_{A}} are important invariants (see for example [5], [14], [17], [20]). In case

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

Received by the editors August 22, 1996 and, in revised form, April 25, 1997.

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

A is a triangular algebra, we denote by

ρ̃(A) = max {ρ(ϕ_{B}): B = A/AeA for some idempotent e ∈ A}.

Theorem 1. Assume that A is a triangular algebra such that

ρ̃(A) ≤ \frac{t − 2 + \sqrt{t^{2} − 4t}}{2}

for some natural number t ≥ 4. Let 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split

sequence in mod_{A} such that Y_{i} is directing (i = 1, …, s(X)). Then s(X) ≤ t + 1.

Theorem 2. Assume A is a tame algebra and let 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0

be an almost split sequence in mod_{A} such that Y_{i} is directing (i = 1, …, s(X)). Then the following holds:

(a) s(X) ≤ 5. Moreover, if s(X) = 5, then for some j ∈ {1, …, s(X)}, the module Y_{j} is projective and injective.

(b) If s(X) ≥ 3, and Y_{1} ≅ Y_{2}, then s(X) = 3 and the module Y_{3} is projective and injective.

Theorem 3. Assume A is a cycle-finite algebra. Then for any almost split se-

quence 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 we have s(X) ≤ 5. If s(X) = 5, there is

some Y_{j} (1 ≤ j ≤ s) which is projective and injective.

We prove the theorems in section 2 after some preliminary considerations. We gratefully acknowledge support of CONACYT and DGAPA, UNAM.

1. Cycles and almost split sequences

1.1. Let H = kΔ be the path algebra of a quiver Δ without oriented cycles (see [6]). A tilting module T in mod_{H} satisfies: Ext_{H}^{1}(T, T) = 0 and there is an exact sequence 0 → H → T′ → T′′ → 0 with T′, T′′ ∈ add T (see [18]). For a tilting module _{H}T, the algebra B = End_{H}(T) is called a tilted algebra of type Δ.

We recall that an indecomposable B-module X is sincere if Hom_{B}(P, X) ≠ 0 for every projective B-module P. If X is a sincere directing indecomposable B-module, then B is a tilted algebra [18, p. 375].

1.2. The Auslander-Reiten quiver Γ_{A} of A has as vertices representatives of the iso-classes of indecomposable modules; there are as many arrows X → Y in Γ_{A} as dim_{k} rad_{A}(X, Y)/ rad_{A}^{2}(X, Y). A path Y_{0} → Y_{1} → ⋯ → Y_{s} in Γ_{A} is said to be sectional if τ_{A}Y_{i+1} ≠ Y_{i−1} for all i = 1, …, s − 1. In case Y_{0} = Y_{s}, we have a sectional cycle if τ_{A}Y_{i+1} ≠ Y_{i−1} for all i = 1, …, s mod s. A sectional path in Γ_{A} contains no sectional cycle [4].

A component C of Γ_{A} is directing if it is formed by directing modules.

1.3. Let η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split sequence in mod_{A}

and let B(η) be a quotient A/AeA, with e idempotent in A, of minimal dimension such that η is formed by B(η)-modules. By [18], one of the modules τ_{A}X, X or Y_{i} (1 ≤ i ≤ s(X)) is sincere as B(η)-module.

Lemma. Let η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split sequence in

mod_{A} such that Y_{i} is directing as B(η)-module, for i = 1, …, s(X). Then

(a) X and τ_{A}X are directing as B(η)-modules;

(b) B(η) is a tilted algebra.

Proof. Consider the almost split sequence η: 0 → τ_{A}X ⟶^{(h_{i})} ⨁_{i=1}^{s(X)} Y_{i} ⟶^{(g_{i})} X → 0.

Let X = Z_{0} ⟶^{f_{1}} Z_{1} ⟶ ⋯ ⟶ Z_{s−1} ⟶^{f_{s}} Z_{s} = X be a cycle in mod_{B(η)}. The

map f_{s}: Z_{s−1} → Z_{s} = X factorizes through the sink map (g_{i})_{i}: ⨁_{i=1}^{s(X)} Y_{i} → X and

for some j ∈ {1, …, s(X)} we get a non-zero map f′_{s}: Z_{s−1} → Y_{j}. Therefore we get a cycle

Y_{j} ⟶^{g_{j}} X = Z_{0} ⟶^{f_{1}} Z_{1} ⟶ ⋯ ⟶ Z_{s−1} ⟶^{f′_{s}} Y_{j}.

A contradiction. Hence X (and similarly τ_{A}X) is directing. The algebra B(η) is tilted by (1.1).

1.4 Proposition. Let X be a directing indecomposable non-projective module and

η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be the corresponding almost split sequence.

Assume s(X) ≥ 3. Then one of the following conditions holds:

(a) B(η) is a tilted algebra and X belongs to a directing component of Γ_{B(η)};

(b) there exists a sectional path τ_{A}X = Z_{0} → Z_{1} → ⋯ → Z_{s} in Γ_{B(η)} with Z_{s} an injective module.

Proof. We shall denote B = B(η). Assume first that Y_{i} is directing as B-module for i = 1, …, s(X). By (1.3), B is a tilted algebra. By [9], we know the structure of Γ_{B}: there is a postprojective, a preinjective and a connecting component (some of these components may coincide) and components of type ℤ𝔸_{∞} or ℤ𝔸_{∞}/(n), possibly with inserted ray modules or coinserted coray modules. Since s(X) ≥ 3, then X belongs to a postprojective, preinjective or connecting component of Γ_{B}, all of which are directing. Hence (a) holds.

Assume Y_{1} is not directing. Let Y_{1} = Z_{0} ⟶^{f_{1}} Z_{1} ⟶ ⋯ ⟶^{f_{s}} Z_{s} = Y_{1} be a cycle in mod_{B}. By (1.2), we may assume that one of the following situations occurs:

(1) Y_{1} = Z_{0} ⟶^{f_{1}} Z_{1} ⟶ ⋯ ⟶^{f_{r}} Z_{r} ⟶^{f_{r+1}} Z_{r+1} ⟶ ⋯ ⟶ Y_{1} such that

Z_{0} ⟶^{f_{1}} Z_{1} ⟶ ⋯ ⟶^{f_{r}} Z_{r} is a sectional path in Γ_{B} and τ_{B}Z_{r+1} = Z_{r−1};

(2) for all i ∈ ℕ, there is a sectional path in Γ_{B}, Y_{1} = Z_{0} ⟶^{f_{1}} Z_{1} ⟶ ⋯ ⟶^{f_{i}} Z_{i} and a path Z_{i} ⟶ Z′_{i+1} ⟶ ⋯ ⟶ Y_{1} in mod_{B}.

If some Z_{i} is injective (1 ≤ i ≤ r in case (1) or 1 ≤ i in case (2)), then (b) holds. We assume that no Z_{i} is injective in order to get a contradiction.

First observe that situation (1) cannot happen. Otherwise, we get a cycle

Y_{1} ⟶^{h_{1}} X ⟶ τ −_{B}Y_{1} ⟶ τ −_{B}Z_{1} ⟶ ⋯ τ −_{B}Z_{r−1} = Z_{r+1} ⟶ ⋯ ⟶ Y_{1} ⟶^{h_{1}} X,

where h_{1} : Y_{1} → X is an irreducible map. Contradicting that X is a directing A-module.

Let n be the number of iso-classes of simple A-modules and consider a sectional path Y_{1} = Z_{0} ⟶^{f_{1}} Z_{1} ⟶^{f_{2}} Z_{2} ⟶ ⋯ ⟶^{f_{n}} Z_{n} in Γ_{B} as given in situation (2). Moreover, there is a path Z_{n} → Z′_{n+1} → ⋯ → Y_{1} in mod_{B}. We shall prove that

⨁_{i=0}^{n} Z_{i} is a partial cotilting module, which yields the desired contradiction. Indeed,

since the number of summands of the partial cotilting module is not bigger than n, then we get Z_{i} ≅ Z_{j} for some j > i. By (1.2), the cycle Z_{i} → ⋯ Z_{j−1} → Z_{j} is not sectional and hence Z_{j−1} ≅ τ_{B}Z_{i+1} or Z_{j−2} ≅ τ_{B}Z_{i} which yields a cycle through X as above.

Let us first show that i dim_{B} Z_{i} ≤ 1, i = 0, …, n. Otherwise i dim_{B} Z_{i} > 1 and there are an indecomposable projective B-module P and a map 0 ≠ g ∈

Hom_{B}(τ −_{B}Z_{i}, P) (see [18]). Since ⨁_{i=1}^{s(X)} Y_{i} is a sincere B-module, there are some

j ∈ {1, …, s(X)} and a map 0 ≠ g′ ∈ Hom_{B}(P, Y_{j}). We get a cycle

X ⟶ τ −_{B}Y_{j} ⟶ ⋯ ⟶ τ −_{B}Z_{i} ⟶^{g} P ⟶^{g′} Y_{j} ⟶^{h_{j}} X,

again a contradiction.

Let i, j ∈ {0, …, n}; we show that Ext_{B}^{1}(Z_{i}, Z_{j}) = 0. Otherwise, there is a map 0 ≠ g ∈ Hom_{B}(τ −_{B}Z_{j}, Z_{i}). Then we get a cycle

X ⟶ τ −_{B}Y_{j} ⟶ ⋯ ⟶ τ −_{B}Z_{j} ⟶^{g} Z_{i} ⟶^{f_{i+1}} Z_{i+1} ⟶ ⋯

⋯ → Z_{n} ⟶ ⋯ ⟶ Y_{1} ⟶^{h_{i}} X,

and a contradiction. This shows that ⨁_{i=0}^{n} Z_{i} is a partial cotilting module, which

completes the proof.

1.5. We say that X is a predecessor of Y in Γ_{A} (and Y a successor of X) if there is a path X = Z_{0} → Z_{1} → ⋯ → Z_{s} = Y in Γ_{A}.

Proposition. Let η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split sequence

such that Y_{1} ≅ Y_{2}. Then

(a) If dim_{k} τ_{A}X < dim_{k} Y_{1}, then τ −n_{A}X and τ −n_{A}Y_{i} (1 ≤ i ≤ s(X)) are well defined for all n ≥ 0. Moreover, X has no injective successors and does not belong to any oriented cycle in Γ_{A}.

(b) If dim_{k} τ_{A}X > dim_{k} Y_{1}, τ n_{A}X and τ n_{A}Y_{1} are well defined for all n ≥ 0. Moreover, τ_{A}X has no projective predecessors and does not belong to any oriented cycle in Γ_{A}.

Proof. (a): Let η: 0 → τ_{A}X ⟶^{(h_{i})} ⨁_{i=1}^{s(X)} Y_{i} ⟶^{(g_{i})} X → 0 be an almost split

sequence with Y_{1} ≅ Y_{2} and assume that h_{1} is a monomorphism. Then for all 1 ≤ i ≤ s(X), dim_{k} Y_{i} < dim_{k} X. In particular, no Y_{i} is injective. Since the almost split sequence starting at Y_{1} has X at least twice as a summand of its middle term, then the above argument implies that X is not injective. Moreover, dim_{k} τ_{A}X < dim_{k} X < dim_{k} τ −_{A}X. We proceed in this way to show that τ −n_{A}X and τ −n_{A}Y_{i} (1 ≤ i ≤ s(X)) are well defined for all n ≥ 0.

Assume that Y is a minimal injective successor of X. Then we find a sectional path τ s_{A}X = Z_{0} ⟶^{f_{1}} Z_{1} ⟶^{f_{2}} ⋯ ⟶^{f_{t}} Z_{t} = Y, for some s ≥ 0. Since there is an irreducible monomorphism Z_{1} → τ −_{A}Z_{0}, then Z_{1} is not injective and there is an irreducible monomorphism Z_{2} → τ −_{A}Z_{1}. Continuing this way, we find a monomorphism Z_{t} → τ −_{A}Z_{t−1}, which is a contradiction.

Let Γ be the right stable component of Γ_{A} containing X. Then Γ contains all the successors of X and it has no periodic modules (since dim_{k} τ −n_{A}X grows with n ≥ 0 and hence X is not periodic). If X belongs to an oriented cycle in Γ_{A}, then the cycle belongs to Γ and by [10], Γ has trivial valuation, a contradiction.

(b): With the notation above, assume that h_{1} is an epimorphism. In particular, Y_{1} is not projective. Since the almost split sequence ending at Y_{1} has τ_{A}X at least twice as a summand of its middle term, then dim_{k} τ_{A}X < dim_{k} τ_{A}Y_{1} and τ_{A}X is not projective. By (a), dim_{k} τ 2_{A}X > dim_{k} τ_{A}Y_{1}. We proceed by induction to show that τ n_{A}X and τ n_{A}Y_{1} are well defined for all n ≥ 0.

If, for some 3 ≤ j ≤ s(X), the module Y_{j} is not projective, then dim_{k} τ 2_{A}X > dim_{k} τ_{A}Y_{j}. By induction, τ n_{A}Y_{j} is well defined. The second part of the statement follows from dual arguments to those used in (a).

1.6. The following results of S. Liu are important in the proof of our theorems.

Proposition [9]. Let 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split sequence in

mod_{A}. Then

(a) Assume there is a sectional path τ_{A}X = Z_{0} → Z_{1} → ⋯ → Z_{t} in Γ_{A} with Z_{t} injective. Assume moreover that s(X) > 4 or that s(X) = 4 but no Y_{i} is injective (i = 1, …, s(X)). Then τ_{A}X has no projective predecessor in Γ_{A} and τ_{A}X does not belong to a finite cycle in mod_{A}.

(b) If τ_{A}X belongs to a finite cycle in mod_{A}, then s(X) ≤ 4 and in case s(X) = 4, there is a Y_{i} (1 ≤ i ≤ s(X)), which is projective.

1.7. The following result generalizes [3] quoted in the introduction.

Theorem [9]. Let η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split sequence

such that τ_{A}X has a projective predecessor and X has an injective successor in Γ_{A}. Then s(X) ≤ 4 and in case s(X) = 4, there is a Y_{i} (1 ≤ i ≤ 4) which is projective and injective.

2. Proof of the main results

2.1. Proof of the Theorem 1 . Let 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 be an almost split

sequence in mod_{A} and assume that Y_{i} is directing (1 ≤ i ≤ s(X)). Assume that ρ̃(A) ≤ \frac{t−1+\sqrt{t^{2}−4t}}{2} for a natural number t ≥ 4. We assume that s = s(X) ≥ 5.

Let B = B(η). By the proof of (1.4), we may assume that B is a tilted algebra and X belongs to a directing component C of Γ_{B}.

By (1.7), we may assume that X has no injective successors in C. Suppose that Y_{s} is injective in mod_{B}. Since τ_{A}X → Y_{s} is a sectional map, by (1.6), we get that τ_{A}X has no projective predecessors in C. Moreover, no Y_{i}, 1 ≤ i ≤ s − 1, is projective. Indeed, if Y_{1} were projective, then

dim_{k} Y_{1} + dim_{k} Y_{s} > dim_{k} τ_{A}X + dim_{k} X = ∑_{i=1}^{s} dim_{k} Y_{i},

a contradiction. By duality, we may assume that one of the following situations holds:

(1) 0 → τ m+1_{B}X → ⨁_{i=1}^{s} τ m_{B}Y_{i} → τ m_{B}X → 0 is a well-defined almost split sequence

in mod_{B}, for all m ≥ 0;

(2) Y_{s} is projective and injective and 0 → τ m+1_{B}X → ⨁_{i=1}^{s−1} τ m_{B}Y_{i} → τ m_{B}X → 0 is a

well-defined almost split sequence in mod_{B}, for all m ≥ 1. In case (1) we shall prove that s ≤ t; in case (2), the same proof yields s − 1 ≤ t

and hence the result.

Assume that Y_{s} is not projective (i.e. case (1)). Consider the Grothendieck group K_{0}(B) and the Coxeter matrix ϕ_{B} as a linear transformation ϕ_{B}: K_{0}(B) → K_{0}(B). Since B is tilted, there are a quiver Δ and a tilting kΔ-module T such that B = End_{kΔ}(T). Therefore, there is an isometry σ: K_{0}(kΔ) → K_{0}(B), [L] ↦ [Hom_{kΔ}(T, L)]−[Ext_{kΔ}^{1}(T, L)], where [Y] denotes the class in K_{0}(B) of a B-module Y. Moreover, ϕ_{B}σ = σϕ_{Δ} (see [18]). In particular, ρ(ϕ_{B}) = ρ(ϕ_{Δ}).

Observe that Δ contains the subquiver Δ(s_{1}, …, s_{m}) formed by one source 0,

sinks 1, …, m and s_{i} arrows from 0 to i such that ∑_{i=1}^{m} s_{i} = s. Indeed, Δ = S^{op},

where S is the slice in the component C formed as the full subquiver of C whose vertices Z are the starting points of sectional paths to X. Since S contains the arrows Y_{i} → X, i = 1, …, s, we get the claim (of course, m is the number of isoclasses among the Y_{i}). Moreover, by [20], we have ρ(ϕ_{Δ}) ≥ ρ(ϕ_{Δ(s_{1},... ,s_{m})}).

On the other hand, ρ(ϕ_{Δ(s_{1},... ,s_{m})})= \frac{b−2+\sqrt{b^{2}−4b}}{2}, where b= ∑_{i=1}^{m} s_{i}^{2}.

Hence ρ(ϕ_{Δ(s_{1},... ,s_{m})})≥ \frac{s−2+\sqrt{s^{2}−4s}}{2} = ρ(ϕ_{Δ_{0}}), where

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Altogether we get

\frac{t − 2 + \sqrt{t^{2} − 4t}}{2} ≥ ρ̃(A) ≥ ρ(ϕ_{B}) = ρ(ϕ_{Δ}) ≥ ρ(ϕ_{Δ_{0}}) = \frac{s − 2 + \sqrt{s^{2} − 4s}}{2}.

Hence t ≥ s, as desired.

2.2. We recall that the algebra A is said to be tame if for every d ∈ ℕ there are finitely many A − k[t]-bimodules M_{1}, …, M_{s} which are finitely generated free as right k[t]-modules and such that every indecomposable A-module X with dim_{k} X = d is isomorphic to M_{i} ⊗_{k[t]} S for some 1 ≤ i ≤ s and some simple k[t]-module S.

From now on we will assume that A is a basic connected algebra of the form A = kQ/I (see [6]). The following result follows an idea of von Höhne (see [12]).

Proposition. Let A = kQ/I be a tame algebra and let X be a non-projective

indecomposable module such that the almost split sequence 0 → τ_{A}X → ⨁_{i=1}^{s} Y_{i} →

X → 0 has directing middle terms Y_{i}, i = 1, …, s. Then for all vertices i ∈ Q_{0} we have

| dim_{k} τ_{A}X(i) − dim_{k} X(i)| ≤ 2.

Proof. Let B = B(η). By (1.3), B is a tame tilted algebra. Then it easily follows that p dim_{B} X ≤ 1 and Hom_{B}(X,_{B}B) = 0. Therefore, by [18], [τ_{A}X] = [τ_{B}X] = ϕ_{B}[X].

Consider the Euler (non-symmetric) bilinear form ⟨−, −⟩: K_{0}(B) × K_{0}(B) → ℤ

defined by ⟨[L], [M]⟩ = ∑_{i=0}^{∞}(−1)^{i} dim_{k} Ext_{B}^{i}(L, M) and the corresponding quadratic

form χ_{B}. Since gℓ dim B ≤ 2, then χ_{B} is also the Tits form of B and therefore χ_{B} is weakly non-negative (see for example [12]). We have χ_{B}([X]) = 1 = χ_{B}([P_{i}]), for the indecomposable projective module P_{i} corresponding to the vertex i ∈ Q_{0}. Then

0 ≤ χ_{B}([X] + [P_{i}]) = 2 + ⟨[P_{i}], [X]⟩ + ⟨[X], [P_{i}]⟩
= 2 + dim_{k} X(i) − ⟨[P_{i}], ϕ_{B}[X]⟩
= 2 + dim_{k} X(i) − dim_{k} τ_{A}X(i).

Hence, dim_{k} τ_{A}X(i) − dim_{k} X(i) ≤ 2 and similarly we get the other inequality.

Corollary. Let A be a tame algebra and C be a directing component of Γ_{A}. Let X ∈ C be such that τ m_{A}X is well-defined for all m ≥ 0. Then lim_{m→∞} \sqrt[m]{dim_{k} τ m_{A}X} = 1.

2.3. Proof of the Theorem 2 . Let A be a tame algebra and consider an almost split

sequence η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0 such that Y_{i} is directing, 1 ≤ i ≤ s(X).

(a): Assume that s = s(X) ≥ 5.

Let B = B(η) be a tilted algebra by (1.4) and assume that X belongs to a directing component C of Γ_{B}. As in (2.1), we may assume that one of the following situations occurs:

(1) 0 → τ m+1_{B}X → ⨁_{i=1}^{s} τ m_{B}Y_{i} → τ m_{B}X → 0 is well-defined for all m ≥ 0;

(2) Y_{s} is projective and injective and 0 → τ m+1_{B}X → ⨁_{i=1}^{s−1} τ m_{B}Y_{i} → τ m_{B}X → 0 is

well-defined for m ≥ 1. Assume that case (1) holds. Consider P′ the direct sum of all indecomposable projective modules in C. Let

P′ = Be′ and C = B/Be′B which is a quotient of A by the two-sided ideal generated by an idempotent element. Observe that for m ≥ 1, τ m_{B}X is a C-module. Indeed, otherwise an indecomposable direct summand P of P′ would be a predecessor of τ m_{B}X, for some m ≥ 1, and hence a predecessor of τ_{B}X, a contradiction to (1.6).

Moreover, C is a tilted algebra. Indeed, consider S the slice in C formed by those Z ∈ C such that there is a sectional path from Z to X (see [18]). We claim that any indecomposable projective P in C belongs to S. Indeed, P is a predecessor of some Y_{j} (1 ≤ j ≤ s) because ⨁_{i=1}^{s} Y_{i} is a sincere B-module. Consider a path

P = Z_{0} → Z_{1} → ⋯ → Z_{t} = Y_{j} → X and assume it is not sectional. Hence for some 1 ≤ i ≤ t−1, τ_{B}Z_{i+1} = Z_{i−1}. We may assume that the modules Z_{i}, 1 ≤ i ≤ t, are not projectives. Then we get a path

P → Z_{1} → ⋯ → Z_{i−1} → τ_{B}Z_{i+2} → τ_{B}Z_{i+3} → ⋯ → τ_{B}Y_{j} → τ_{B}X,

a contradiction. Therefore, τ_{B}S is a slice in Γ_{C} and C is a tilted algebra.

We shall consider N = τ_{B}X as C-module. Observe that τ m_{C}N = τ m+1_{B}X is welldefined. Consider the Grothendieck group K_{0}(C) of C and the Coxeter matrix ϕ_{C} as a linear transformation ϕ_{C}: K_{0}(C) → K_{0}(C). We want to show that for m ≥ 0, the class [τ m_{C}N] ∈ K_{0}(C) satisfies [τ m_{C}N] = ϕ_{C}^{m}[N]. By [18], it is enough to show that p dim_{C} τ m_{C}N ≤ 1 and Hom_{C}(τ m_{C}N,_{C}C) = 0 for m ≥ 0.

Indeed, if p dim_{C} τ m_{C}N > 1, there is a map 0 ≠ g ∈ Hom_{C}(I, τ m+1_{C}N) for some indecomposable injective C-module I. As B-module, I is also injective (otherwise, τ −_{B}I is a predecessor of τ m_{C}N = τ m+1_{B}X and τ −_{B}I ∈ mod_{C}). Hence there is a map

0 ≠ g′ ∈ Hom_{B}(Y_{j}, I) for some 1 ≤ j ≤ s, because ⨁_{i=1}^{s} Y_{i} is sincere. We get a cycle,

Y_{j} ⟶^{g′} I ⟶^{g} τ m+1_{C}N = τ m+2_{B}X ⟶ ⋯ ⟶ τ_{B}X ⟶ Y_{j},

a contradiction. Now, assume 0 ≠ g ∈ Hom_{C}(τ m_{C}N, P̄) for some indecomposable projective C-module P̄. Since τ_{B}S is a slice in Γ_{C}, then P̄ is a predecessor of N and as B-module, P̄ ∈ C (use convexity of C, see [15]). Therefore P̄ is also projective as B-module (otherwise τ_{B} P̄ is defined and Hom_{B}(P′, τ_{B} P̄) = 0, for any projective P′ in S; hence τ_{B} P̄ ∈ mod_{C}, a contradiction). This contradicts (1.6).

Let Δ be a quiver such that C = End_{kΔ}(T) for a tilting module T. Consider the isometry σ: K_{0}(kΔ) → K_{0}(C), [L] ↦ [Hom_{kΔ}(T, L)] − [Ext_{kΔ}^{1}(T, L)], which satisfies ϕ_{C}σ = σϕ_{Δ}. Observe that T does not have preinjective direct summands (since there are no projective modules in the connecting component of Γ_{C}) and since N is a predecessor of a slice in Γ_{C}, then N = ΣL for some preinjective kΔmodule L, with Σ = Hom_{kΔ}(T, −). Hence [18] implies that τ m_{C}N = Στ m_{kΔ}L and

[τ m_{C}N] = σϕ_{Δ}^{m}[L], for m ≥ 0.

We have the following facts:

(i) lim_{m→∞} \sqrt[m]{dim_{k} τ m_{C}N} exists and equals lim_{m→∞} \sqrt[m]{dim_{k} τ_{kΔ}^{m}L} (see [12] for a detailed proof).

(ii) lim_{m→∞} \sqrt[m]{dim_{k} τ_{kΔ}^{m}L} = ρ(ϕ_{Δ}) (see [5], [15]).

(iii) Δ contains a subquiver Δ(s_{1}, …, s_{m}) with s = ∑_{i=1}^{m} s_{i} (notation as in 2.1).

By (2.2), 1 = lim_{m→∞} \sqrt[m]{dim_{k} τ m_{C}N} = ρ(ϕ_{Δ}) ≥ ρ(ϕ_{Δ(s_{1},... ,s_{m})}) ≥ \frac{s−2+\sqrt{s^{2}−4s}}{2}.

Therefore s ≤ 4, contradicting our assumption on s(X).

We get that situation (2) holds, that is, Y_{s} is projective and injective and for all

m ≥ 0, 0 → τ m+1_{B}X → ⨁_{i=1}^{s−1} τ m_{B}Y_{i} → τ m_{B}X → 0 is well defined. Proceeding as above

we get that s − 1 ≤ 4. Hence s(X) = 5 and Y_{s} is projective and injective.

(b): Assume that s(X) ≥ 3 and that Y_{1} ≅ Y_{2}. Then by (1.5), we may assume that either case (1) or case (2) in part (a) of the proof holds. In case (1), the argument above yields 1 ≥ ρ(ϕ_{Δ(s_{1},... ,s_{m})}), with ∑_{i=1}^{m} s_{i} = s(X) and s_{1} ≥ 2, which is impossible since ρ(ϕ_{Δ(2,1)}) and ρ(ϕ_{Δ(3)}) are both strictly bigger than 1. Hence, case (2) holds and 2 = s_{1} = s(X) − 1.

2.4. Proof of Theorem 3 . Assume A is a cycle-finite algebra, hence tame. Let X

be an indecomposable non-projective module and η: 0 → τ_{A}X → ⨁_{i=1}^{s(X)} Y_{i} → X → 0

an almost split sequence. Assume that s = s(X) ≥ 5. Let B = B(η).

If X is not directing, then X belongs to a finite cycle in mod_{A}. By (1.6), we should have s ≤ 4. Then X and τ_{A}X are directing. In case all Y_{i} (1 ≤ i ≤ s) are directing, Theorem 2 shows that s = 5.

We may assume that Y_{1} is not directing. By (1.4), there is a sectional path τ_{A}X = Z_{0} → Z_{1} → ⋯ → Z_{s} in Γ_{B} with Z_{s} an injective B-module. By (1.6), τ_{A}X has no projective predecessor in Γ_{B}.

Assume that no Y_{i} is a projective B-module, 1 ≤ i ≤ s. Then 0 → τ m+1_{B}X →

⨁_{i=1}^{s} τ m_{B}Y_{i} → τ m_{B}X → 0 is well-defined for all m ≥ 0. Consider the sequence τη: 0 →

τ 2_{B}X → ⨁_{i=1}^{s} τ_{B}Y_{i} → τ_{B}X → 0. If all τ_{B}Y_{i} are directing as B-modules (1 ≤ i ≤ s),

then by Theorem 2 we are done. Hence by (1.4), we consider the quotient algebra B_{1} = B(τη) and we get a sectional path Z′_{t} → ⋯ → Z′_{1} → Z′_{0} = τ_{B}X in Γ_{B_{1}} with Z′_{t} a projective B_{1}-module. Since Z′_{t} cannot be a projective B-module predecessor of τ_{A}X, then B_{1} is a proper quotient of B. Consider the sequence τ^{2}η: 0 → τ 3_{B}X →

⨁_{i=1}^{s} τ 2_{B}Y_{i} → τ 2_{B}X → 0. Again either all τ 2_{B}Y_{i} are directing as B_{1}-modules and s ≤ 5

or B_{2} = B(τ^{2}η) is a proper quotient of B_{1}. Repeating the process finitely many steps we end with an algebra B_{t} = B(τ^{t}η) where all τ t_{B}Y_{i} are directing. Theorem 2 yields s = 5 and some τ t_{B}Y_{j} is projective (1 ≤ j ≤ s), a contradiction.

We may therefore assume that Y_{s} is a projective B-module. By the dual argument, we may suppose that some Y_{j} (1 ≤ j ≤ s) is an injective B-module. As in (2.1), j = s. Let Y_{s} = P_{w} be a projective-injective B-module. Then the irreducible maps of η look as follows:

................................................................................................................................... ................................................................................................................................................ ......................................................................................... ............ ......................................................................................................... ............ ..................................................................................................... ............  ............................................................................................................................................. ............ .................................................................................................................... ............ .................................................................................................................... ................................................................................................................................._{...................}  ^{.................}_{...................} τ_{A}X  Y_{1} ⋮ Y_{s−1} Y_{s} = P_{w}  X

Since τ_{A}X is directing in mod_{A}, then τ_{A}X(w) = 0. Hence Y_{i}(w) = 0 for 1 ≤

i ≤ s − 1. Then Y_{s} = P_{w} is a sincere B-module and B = C[M] = (\begin{matrix}C & M \\ 0 & k \end{matrix}) is

a one-point extension of an algebra C by the module M = rad P_{w}. Since P_{w} is injective, also M = τ_{A}X. Hence rad P_{w} is directing as A-module, and therefore P_{w} is directing as B-module. We get that B is a tilted algebra and by Theorem 2 we are done.

3866 J. A. DE LA PEÑA AND M. TAKANE

2.5 Examples. a) The bound in Theorems 1 and 2 is optimal. Consider A_{t} the algebra given by the quiver Q

..................................................................................................................................................................................................... ....................................................................................................................................................................... .....................................................................................................................................................................................................  ................................................................................................................................................................................................. ............................................................................................................................................................... ........ ................................................................................................................................................................................................. w  α_{1} α_{2} ⋮ α_{t}  1 2t  β_{1} β_{t}  0

and ideal I_{t} generated by ∑_{i=1}^{t} β_{i}α_{i}. Then ρ̃(A_{t}) = \frac{t−2+\sqrt{t^{2}−4}}{2}. There is an almost

split sequence as follows:

................................................................................................................................ ............................................................................................ ................................................................................................................................................  ................................................................................................................................ ............................................................................................ ................................................................................................................................................ τ −_{A}P_{0}  τ −_{A}P_{1} ⋮ τ −_{A}P_{t} P_{w}  τ −2_{A}P_{0}

that is, s(τ −2_{A}P_{0}) = t + 1. The algebra A_{t} is tame if and only if t ≤ 3.

b) Consider B_{t} the algebra given by the above quiver Q and ideal I′_{t} generated by β_{i}α_{i} − β_{1}α_{1}, i = 2, …, t. Again ρ̃(B_{t}) = \frac{t−2+\sqrt{t^{2}−4}}{2}. There is an almost split sequence as follows:

....................................................................................................................... ....................................................................................... .......................................................................................................  ............................................................................................................... .................................................................................................................. ........ ................................................................................................................................. rad P_{w}  P_{w} = I_{0} S_{1} ⋮ S_{t}  I_{0}/S_{0}

That is, s(I_{0}/S_{0}) = t + 1. The algebra B_{t} is tame for t ≤ 4.

THE MIDDLE OF AN ALMOST SPLIT SEQUENCE 3867

c) Consider the algebra C given by the quiver

4^{..............................................................................................................................................} α_{4} β_{4}_{............................................................................................................................................}w^{................................................................................................................................... .........} β_{2} α_{2}_{............................................................................................................................................} 2^{................................................................................................................................................} β_{1}  α_{1}^{...................................................................................................................................... .........} 1_{..................................................................................................................... ................ .........} α_{3}  β_{3}_{..................................................................................................................  ............. .........} 3

with ideal I generated by β_{i}α_{i} − β_{1}α_{1}, i = 2, 3, 4 and α_{j}β_{i} = 0 for any 1 ≤ i, j ≤ 4. Then C has a Galois covering C̃ given by the quiver Q̃

................................................................................................................................ .................................................................................................. ........ .......................................................................................................... ................................................................................................................................ .................................................................................................................................... ........................................................................................................ ................................................................................................ ........ ....................................................................................................................................  ................................................................................................................................ .................................................................................................. ........ .......................................................................................................... ................................................................................................................................ .................................................................................................................................... ........................................................................................................ ................................................................................................ ........ ....................................................................................................................................  ................................................................................................................................ .................................................................................................. ........ .......................................................................................................... ................................................................................................................................ .................................................................................................................................... ........................................................................................................ ................................................................................................ ........ .................................................................................................................................... · · · • • • •  •  • • • •  •  • • • •  •  • • • • ⋯ β_{01} β_{04}  α_{11} α_{14}  β_{11} β_{14}  α_{21} α_{24}

and ideal Ĩ generated by β_{ji}α_{ji} − β_{j1}α_{j1} for i = 2, 3, 4 and j ∈ ℤ and α_{j+1,s}β_{j,t} = 0 for all 1 ≤ s, t ≤ 4 and j ∈ ℤ.

It is easy to check that C̃ has no hypercritical or pg-critical subcategories and hence C̃ is tame of polynomial growth [19]. Since the covering π: C̃ → C is defined by the action of the free group ℤ, then C is tame [13], [19]. There is an almost split sequence in mod_{C} of the form

............................................................................................................................................................................... ................................................................................................................................................. ............................................................................................................................. ......................................................................................................................................... ........ .....................................................................................................................................................................................................  ............................................................................................................................................................................... ......................................................................................................................................... ........ ............................................................................................................................. ................................................................................................................................................. ..................................................................................................................................................................................................... rad P_{w}  P_{w} = I_{w} S_{1} S_{2} S_{3} S_{4}  I_{w}/S_{w}

and there are cycles in mod_{C} as follows:

S_{i} → I_{i} → I_{w} = P_{w} → P_{i} → S_{i}

for all i = 1, …, 4.

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Instituto de Matemáticas, UNAM Ciudad Universitaria 04510 México, D. F. México

E-mail address: jap@penelope.matem.unam.mx
E-mail address: takane@gauss.matem.unam.mx